Properties

Label 3.65.b.a
Level 3
Weight 65
Character orbit 3.b
Analytic conductor 77.821
Analytic rank 0
Dimension 20
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 65 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(77.8210007872\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{190}\cdot 3^{282}\cdot 5^{19}\cdot 7^{8}\cdot 11^{4}\cdot 13^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +\beta_{1} q^{2} +(-71037898477915 + 13312 \beta_{1} + \beta_{2}) q^{3} +(-7875089044249319489 - 34 \beta_{1} - 68 \beta_{2} + \beta_{3}) q^{4} +(199423 - 628811347993 \beta_{1} - 498558 \beta_{2} + \beta_{4}) q^{5} +(-\)\(35\!\cdots\!94\)\( - 62197449956605 \beta_{1} - 906631 \beta_{2} + 50602 \beta_{3} + 10 \beta_{4} - \beta_{5}) q^{6} +(\)\(34\!\cdots\!44\)\( - 9140540750 \beta_{1} - 18444142618 \beta_{2} - 3778940 \beta_{3} - 52 \beta_{4} + 11 \beta_{5} + \beta_{6}) q^{7} +(1917597342933 - 6153246468640758853 \beta_{1} - 4793992141335 \beta_{2} - 2512797 \beta_{3} + 80800 \beta_{4} - 22 \beta_{5} - 5 \beta_{7} + \beta_{8}) q^{8} +(-\)\(11\!\cdots\!52\)\( + 6982777574264106921 \beta_{1} - 74329154212246 \beta_{2} + 6516874849 \beta_{3} - 3147010 \beta_{4} - 11598 \beta_{5} - 685 \beta_{6} - 69 \beta_{7} + 4 \beta_{8} + \beta_{9}) q^{9} +O(q^{10})\) \( q +\beta_{1} q^{2} +(-71037898477915 + 13312 \beta_{1} + \beta_{2}) q^{3} +(-7875089044249319489 - 34 \beta_{1} - 68 \beta_{2} + \beta_{3}) q^{4} +(199423 - 628811347993 \beta_{1} - 498558 \beta_{2} + \beta_{4}) q^{5} +(-\)\(35\!\cdots\!94\)\( - 62197449956605 \beta_{1} - 906631 \beta_{2} + 50602 \beta_{3} + 10 \beta_{4} - \beta_{5}) q^{6} +(\)\(34\!\cdots\!44\)\( - 9140540750 \beta_{1} - 18444142618 \beta_{2} - 3778940 \beta_{3} - 52 \beta_{4} + 11 \beta_{5} + \beta_{6}) q^{7} +(1917597342933 - 6153246468640758853 \beta_{1} - 4793992141335 \beta_{2} - 2512797 \beta_{3} + 80800 \beta_{4} - 22 \beta_{5} - 5 \beta_{7} + \beta_{8}) q^{8} +(-\)\(11\!\cdots\!52\)\( + 6982777574264106921 \beta_{1} - 74329154212246 \beta_{2} + 6516874849 \beta_{3} - 3147010 \beta_{4} - 11598 \beta_{5} - 685 \beta_{6} - 69 \beta_{7} + 4 \beta_{8} + \beta_{9}) q^{9} +(\)\(16\!\cdots\!80\)\( - 854010031244499 \beta_{1} - 1723497650112045 \beta_{2} - 758703259719 \beta_{3} - 4857748 \beta_{4} + 2699204 \beta_{5} + 17826 \beta_{6} + 1965 \beta_{7} + 2 \beta_{8} - 7 \beta_{9} - \beta_{10}) q^{10} +(-16207640388503565 + \)\(15\!\cdots\!66\)\( \beta_{1} + 40519094472992719 \beta_{2} + 19672623336 \beta_{3} - 6676152078 \beta_{4} - 42010184 \beta_{5} + 32583 \beta_{6} + 37463 \beta_{7} - 2020 \beta_{8} - 11 \beta_{9} - \beta_{10} + \beta_{11}) q^{11} +(\)\(32\!\cdots\!21\)\( - \)\(95\!\cdots\!00\)\( \beta_{1} - 7793315695710502978 \beta_{2} - 81249096595734 \beta_{3} + 330417294540 \beta_{4} + 17875115 \beta_{5} + 1952173 \beta_{6} - 719618 \beta_{7} + 51961 \beta_{8} + 41 \beta_{9} + 9 \beta_{10} - \beta_{11} - \beta_{14}) q^{12} +(-\)\(37\!\cdots\!56\)\( + 2934646771215608058 \beta_{1} + 5921365666571381287 \beta_{2} + 745123119792968 \beta_{3} + 16644667299 \beta_{4} - 5108668245 \beta_{5} + 53620239 \beta_{6} - 8983119 \beta_{7} - 1822 \beta_{8} - 10061 \beta_{9} + 485 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} + \beta_{13}) q^{13} +(-\)\(12\!\cdots\!69\)\( + \)\(97\!\cdots\!40\)\( \beta_{1} + \)\(31\!\cdots\!16\)\( \beta_{2} + 153339885807252 \beta_{3} - 42947480520127 \beta_{4} + 44366776045 \beta_{5} - 34584154 \beta_{6} + 331024304 \beta_{7} - 11818509 \beta_{8} - 212369 \beta_{9} + 1662 \beta_{10} + 336 \beta_{11} + 49 \beta_{12} + 9 \beta_{14} - \beta_{16}) q^{14} +(\)\(19\!\cdots\!64\)\( + \)\(26\!\cdots\!58\)\( \beta_{1} + 12492511355691608407 \beta_{2} - 482895038272979454 \beta_{3} - 247094460553336 \beta_{4} + 485813030151 \beta_{5} + 8740062865 \beta_{6} - 987078388 \beta_{7} + 27822643 \beta_{8} + 159085 \beta_{9} - 27111 \beta_{10} + 1224 \beta_{11} - 690 \beta_{12} + 14 \beta_{13} + 7 \beta_{14} + \beta_{18}) q^{15} +(\)\(16\!\cdots\!31\)\( - \)\(74\!\cdots\!67\)\( \beta_{1} - \)\(15\!\cdots\!54\)\( \beta_{2} - 3049740477697850010 \beta_{3} - 4532452580139 \beta_{4} + 5127411236751 \beta_{5} + 45368758878 \beta_{6} + 1441158400 \beta_{7} + 2121544 \beta_{8} + 298 \beta_{9} - 173175 \beta_{10} + 3509 \beta_{11} - 3498 \beta_{12} - 51 \beta_{13} - 384 \beta_{14} - \beta_{15} - 2 \beta_{16} + 4 \beta_{18} + \beta_{19}) q^{16} +(-\)\(52\!\cdots\!40\)\( + \)\(14\!\cdots\!81\)\( \beta_{1} + \)\(13\!\cdots\!28\)\( \beta_{2} + 72557907830481327 \beta_{3} + 12238359157350630 \beta_{4} + 27016255914383 \beta_{5} - 21073337386 \beta_{6} + 157678949172 \beta_{7} - 4592687247 \beta_{8} - 58435841 \beta_{9} + 724454 \beta_{10} + 12312 \beta_{11} + 22835 \beta_{12} - 12 \beta_{13} + 8367 \beta_{14} - 15 \beta_{15} - 135 \beta_{16} + \beta_{17} - 30 \beta_{18} + 15 \beta_{19}) q^{17} +(-\)\(18\!\cdots\!94\)\( - \)\(22\!\cdots\!78\)\( \beta_{1} - \)\(33\!\cdots\!50\)\( \beta_{2} - 59452954548818528099 \beta_{3} + 152993153852807080 \beta_{4} + 63013708156336 \beta_{5} + 1955478484920 \beta_{6} + 77046408404 \beta_{7} + 11826922370 \beta_{8} + 174760929 \beta_{9} + 1320869 \beta_{10} - 1395127 \beta_{11} - 44339 \beta_{12} + 14885 \beta_{13} - 14297 \beta_{14} - 82 \beta_{15} + 1998 \beta_{16} + 27 \beta_{17} - 36 \beta_{18} + 81 \beta_{19}) q^{18} +(\)\(54\!\cdots\!76\)\( - \)\(29\!\cdots\!66\)\( \beta_{1} - \)\(58\!\cdots\!86\)\( \beta_{2} - \)\(53\!\cdots\!49\)\( \beta_{3} - 16729038793992343 \beta_{4} + 1190369661730564 \beta_{5} + 2567677875188 \beta_{6} + 7266483515674 \beta_{7} + 381498691 \beta_{8} + 1745933206 \beta_{9} - 19663660 \beta_{10} + 824814 \beta_{11} - 771769 \beta_{12} - 98291 \beta_{13} - 805872 \beta_{14} - 160 \beta_{15} - 593 \beta_{16} + 351 \beta_{17} + 484 \beta_{18} + 121 \beta_{19}) q^{19} +(-\)\(25\!\cdots\!51\)\( + \)\(17\!\cdots\!10\)\( \beta_{1} + \)\(64\!\cdots\!42\)\( \beta_{2} + 32158081066306329304 \beta_{3} - 7763930396012660563 \beta_{4} - 917817487775421 \beta_{5} + 630314319744 \beta_{6} + 69054948047495 \beta_{7} - 979529873927 \beta_{8} - 4705791254 \beta_{9} - 17629461 \beta_{10} + 79833451 \beta_{11} + 658877 \beta_{12} + 9264 \beta_{13} + 6450348 \beta_{14} + 411 \beta_{15} - 38239 \beta_{16} + 2951 \beta_{17} + 2352 \beta_{18} - 1176 \beta_{19}) q^{20} +(-\)\(65\!\cdots\!49\)\( + \)\(41\!\cdots\!22\)\( \beta_{1} + \)\(28\!\cdots\!87\)\( \beta_{2} + \)\(59\!\cdots\!22\)\( \beta_{3} - 19863741218123835450 \beta_{4} - 310315088980465 \beta_{5} + 59664481476301 \beta_{6} + 298632952614196 \beta_{7} - 1032487102835 \beta_{8} - 13155470761 \beta_{9} - 777489231 \beta_{10} - 301999936 \beta_{11} + 16126761 \beta_{12} + 2128812 \beta_{13} + 515666 \beta_{14} - 1596 \beta_{15} - 53055 \beta_{16} + 17577 \beta_{17} - 1890 \beta_{18} - 8505 \beta_{19}) q^{21} +(-\)\(40\!\cdots\!69\)\( - \)\(53\!\cdots\!95\)\( \beta_{1} - \)\(10\!\cdots\!23\)\( \beta_{2} + \)\(48\!\cdots\!24\)\( \beta_{3} - 3066155419452356527 \beta_{4} - 92754783901231163 \beta_{5} - 1245951550267848 \beta_{6} + 1338279594728018 \beta_{7} - 85226958868 \beta_{8} + 506567554531 \beta_{9} + 26578034210 \beta_{10} - 32217005 \beta_{11} + 49163763 \beta_{12} - 4816690 \beta_{13} - 155487780 \beta_{14} - 81188 \beta_{15} - 36118 \beta_{16} + 75114 \beta_{17} - 77992 \beta_{18} - 19498 \beta_{19}) q^{22} +(-\)\(14\!\cdots\!54\)\( + \)\(56\!\cdots\!53\)\( \beta_{1} + \)\(36\!\cdots\!10\)\( \beta_{2} + \)\(18\!\cdots\!10\)\( \beta_{3} - \)\(17\!\cdots\!12\)\( \beta_{4} - 446078378051163656 \beta_{5} + 340656281669016 \beta_{6} + 4046125799079563 \beta_{7} - 32385527067469 \beta_{8} - 391268616874 \beta_{9} - 8747997808 \beta_{10} + 10228351016 \beta_{11} - 130117735 \beta_{12} - 203829 \beta_{13} + 442413822 \beta_{14} - 838596 \beta_{15} + 3421589 \beta_{16} + 211589 \beta_{17} - 68550 \beta_{18} + 34275 \beta_{19}) q^{23} +(\)\(18\!\cdots\!20\)\( + \)\(47\!\cdots\!40\)\( \beta_{1} + \)\(33\!\cdots\!79\)\( \beta_{2} - \)\(11\!\cdots\!31\)\( \beta_{3} + \)\(24\!\cdots\!77\)\( \beta_{4} + 5899090651498748513 \beta_{5} + 7719541392676838 \beta_{6} + 9627490547524699 \beta_{7} - 165445745648527 \beta_{8} - 3687090432070 \beta_{9} - 700936584639 \beta_{10} - 38246151339 \beta_{11} - 577141158 \beta_{12} - 137779679 \beta_{13} + 33951608 \beta_{14} - 5764401 \beta_{15} - 5933574 \beta_{16} + 193428 \beta_{17} + 189284 \beta_{18} + 429381 \beta_{19}) q^{24} +(-\)\(70\!\cdots\!05\)\( + \)\(15\!\cdots\!90\)\( \beta_{1} + \)\(31\!\cdots\!60\)\( \beta_{2} + \)\(57\!\cdots\!55\)\( \beta_{3} + 90415188292277006160 \beta_{4} - 7294231588172849045 \beta_{5} - 53146929860898150 \beta_{6} - 38443289504315095 \beta_{7} - 9746485701530 \beta_{8} + 66827777442335 \beta_{9} + 3402659764620 \beta_{10} + 3413704360 \beta_{11} - 1649585370 \beta_{12} + 554792005 \beta_{13} + 2975579715 \beta_{14} - 31181465 \beta_{15} - 644140 \beta_{16} - 1648620 \beta_{17} + 4585520 \beta_{18} + 1146380 \beta_{19}) q^{25} +(\)\(54\!\cdots\!00\)\( - \)\(49\!\cdots\!16\)\( \beta_{1} - \)\(13\!\cdots\!30\)\( \beta_{2} - \)\(69\!\cdots\!68\)\( \beta_{3} + \)\(10\!\cdots\!28\)\( \beta_{4} - 1942491354017835068 \beta_{5} + 1758611241714284 \beta_{6} - 149189495568022376 \beta_{7} + 1607385692290066 \beta_{8} - 244411595129056 \beta_{9} + 1142820648560 \beta_{10} + 185087261938 \beta_{11} + 6172202872 \beta_{12} - 177870456 \beta_{13} - 23329424142 \beta_{14} - 145371096 \beta_{15} - 141559042 \beta_{16} - 10833120 \beta_{17} + 131760 \beta_{18} - 65880 \beta_{19}) q^{26} +(\)\(25\!\cdots\!92\)\( - \)\(26\!\cdots\!39\)\( \beta_{1} - \)\(10\!\cdots\!52\)\( \beta_{2} + \)\(54\!\cdots\!50\)\( \beta_{3} + \)\(31\!\cdots\!94\)\( \beta_{4} - 4650797471679561366 \beta_{5} - 136496381497549209 \beta_{6} - 626671509104162178 \beta_{7} - 10864634960315547 \beta_{8} - 78066531755163 \beta_{9} - 28993045984011 \beta_{10} + 195305917923 \beta_{11} - 2579955993 \beta_{12} + 1472753493 \beta_{13} - 7417404702 \beta_{14} - 595739376 \beta_{15} + 421127019 \beta_{16} - 35079237 \beta_{17} - 7565886 \beta_{18} - 13829859 \beta_{19}) q^{27} +(-\)\(19\!\cdots\!55\)\( + \)\(43\!\cdots\!94\)\( \beta_{1} + \)\(87\!\cdots\!70\)\( \beta_{2} + \)\(23\!\cdots\!24\)\( \beta_{3} + \)\(25\!\cdots\!11\)\( \beta_{4} - \)\(37\!\cdots\!57\)\( \beta_{5} - 10732729987751067538 \beta_{6} - 868507791090711779 \beta_{7} - 535070300517851 \beta_{8} + 4599190954954652 \beta_{9} + 80728509273195 \beta_{10} + 87771061959 \beta_{11} + 49943292539 \beta_{12} - 22810963336 \beta_{13} + 123439976310 \beta_{14} - 2168751403 \beta_{15} + 137070367 \beta_{16} - 54881631 \beta_{17} - 164377472 \beta_{18} - 41094368 \beta_{19}) q^{28} +(\)\(47\!\cdots\!49\)\( - \)\(22\!\cdots\!51\)\( \beta_{1} - \)\(11\!\cdots\!02\)\( \beta_{2} - \)\(15\!\cdots\!66\)\( \beta_{3} + \)\(18\!\cdots\!23\)\( \beta_{4} + \)\(61\!\cdots\!34\)\( \beta_{5} - 482974434238427300 \beta_{6} + 149248738050676170 \beta_{7} + 275999584544547360 \beta_{8} - 8506922079693366 \beta_{9} + 59345053702844 \beta_{10} - 14062410547328 \beta_{11} - 180238010320 \beta_{12} - 6577555194 \beta_{13} + 105127475562 \beta_{14} - 6825823974 \beta_{15} + 3646751332 \beta_{16} + 82756260 \beta_{17} + 60608520 \beta_{18} - 30304260 \beta_{19}) q^{29} +(-\)\(70\!\cdots\!57\)\( + \)\(10\!\cdots\!76\)\( \beta_{1} + \)\(15\!\cdots\!94\)\( \beta_{2} - \)\(13\!\cdots\!32\)\( \beta_{3} - \)\(11\!\cdots\!83\)\( \beta_{4} - \)\(26\!\cdots\!13\)\( \beta_{5} + 10043449628349885662 \beta_{6} + 30029347211554755122 \beta_{7} - 1587392837162405743 \beta_{8} + 3169787828158855 \beta_{9} + 506524013578230 \beta_{10} + 38056023923980 \beta_{11} + 773710975047 \beta_{12} + 16706520258 \beta_{13} + 44807491765 \beta_{14} - 17987143404 \beta_{15} - 13437920979 \beta_{16} + 816901686 \beta_{17} + 193113288 \beta_{18} + 317610234 \beta_{19}) q^{30} +(\)\(22\!\cdots\!52\)\( - \)\(19\!\cdots\!31\)\( \beta_{1} - \)\(40\!\cdots\!08\)\( \beta_{2} - \)\(40\!\cdots\!84\)\( \beta_{3} - \)\(11\!\cdots\!30\)\( \beta_{4} + \)\(66\!\cdots\!61\)\( \beta_{5} - 51115582406443596003 \beta_{6} + 51033649923839644847 \beta_{7} - 340951742443747 \beta_{8} + 47131026401544338 \beta_{9} - 1693173952424168 \beta_{10} + 3044990106114 \beta_{11} - 370461360911 \beta_{12} + 336491753483 \beta_{13} - 6263029198140 \beta_{14} - 35812218908 \beta_{15} - 4713960223 \beta_{16} + 2630186289 \beta_{17} + 4167547868 \beta_{18} + 1041886967 \beta_{19}) q^{31} +(-\)\(23\!\cdots\!24\)\( - \)\(45\!\cdots\!96\)\( \beta_{1} + \)\(59\!\cdots\!84\)\( \beta_{2} + \)\(29\!\cdots\!40\)\( \beta_{3} - \)\(62\!\cdots\!12\)\( \beta_{4} + \)\(88\!\cdots\!96\)\( \beta_{5} - 7180405238982916768 \beta_{6} + 69408202340896827184 \beta_{7} + 8595418932916343520 \beta_{8} - 63817107993636848 \beta_{9} + 413534403701528 \beta_{10} + 161073025588584 \beta_{11} + 11416942113560 \beta_{12} - 25043452752 \beta_{13} + 9486654356400 \beta_{14} - 36875150376 \beta_{15} - 61336303896 \beta_{16} + 3943899208 \beta_{17} - 2594252640 \beta_{18} + 1297126320 \beta_{19}) q^{32} +(-\)\(14\!\cdots\!67\)\( + \)\(13\!\cdots\!22\)\( \beta_{1} - \)\(17\!\cdots\!36\)\( \beta_{2} + \)\(12\!\cdots\!94\)\( \beta_{3} + \)\(11\!\cdots\!50\)\( \beta_{4} - \)\(13\!\cdots\!81\)\( \beta_{5} + \)\(88\!\cdots\!01\)\( \beta_{6} - 93163434526957835735 \beta_{7} - 22839121816569903721 \beta_{8} + 168989613956335244 \beta_{9} + 13462061503795464 \beta_{10} - 838881391494528 \beta_{11} + 12579378880029 \beta_{12} - 2071625719796 \beta_{13} - 2103308910445 \beta_{14} + 87146435313 \beta_{15} + 273882734163 \beta_{16} - 4121557749 \beta_{17} - 3550049386 \beta_{18} - 5495647419 \beta_{19}) q^{33} +(-\)\(39\!\cdots\!04\)\( + \)\(33\!\cdots\!26\)\( \beta_{1} + \)\(67\!\cdots\!64\)\( \beta_{2} + \)\(79\!\cdots\!20\)\( \beta_{3} + \)\(19\!\cdots\!28\)\( \beta_{4} - \)\(24\!\cdots\!88\)\( \beta_{5} - \)\(42\!\cdots\!60\)\( \beta_{6} - \)\(75\!\cdots\!86\)\( \beta_{7} + 19690443172954746 \beta_{8} - 1207862427507659048 \beta_{9} - 15807473802289388 \beta_{10} - 212999627744960 \beta_{11} + 166935765150022 \beta_{12} - 2567282177394 \beta_{13} + 101847797634012 \beta_{14} + 625273454108 \beta_{15} + 82212462442 \beta_{16} - 42129297558 \beta_{17} - 80166329768 \beta_{18} - 20041582442 \beta_{19}) q^{34} +(\)\(14\!\cdots\!81\)\( - \)\(10\!\cdots\!63\)\( \beta_{1} - \)\(35\!\cdots\!79\)\( \beta_{2} - \)\(18\!\cdots\!57\)\( \beta_{3} + \)\(13\!\cdots\!05\)\( \beta_{4} + \)\(17\!\cdots\!98\)\( \beta_{5} - \)\(13\!\cdots\!57\)\( \beta_{6} - \)\(39\!\cdots\!30\)\( \beta_{7} + \)\(17\!\cdots\!46\)\( \beta_{8} + 2948883940800068337 \beta_{9} - 13043919594859177 \beta_{10} + 1200971684489017 \beta_{11} + 349684346627294 \beta_{12} + 1766395996218 \beta_{13} - 270830186961084 \beta_{14} + 2127390835752 \beta_{15} + 607537326502 \beta_{16} - 120331613178 \beta_{17} + 65845331244 \beta_{18} - 32922665622 \beta_{19}) q^{35} +(\)\(37\!\cdots\!36\)\( + \)\(93\!\cdots\!88\)\( \beta_{1} + \)\(31\!\cdots\!98\)\( \beta_{2} - \)\(30\!\cdots\!67\)\( \beta_{3} - \)\(12\!\cdots\!91\)\( \beta_{4} + \)\(83\!\cdots\!15\)\( \beta_{5} + \)\(45\!\cdots\!28\)\( \beta_{6} - \)\(14\!\cdots\!41\)\( \beta_{7} - \)\(12\!\cdots\!03\)\( \beta_{8} - 4344931943614645994 \beta_{9} - 217073976775206561 \beta_{10} + 1970379144733095 \beta_{11} + 1718683220545317 \beta_{12} + 52983370550448 \beta_{13} + 58631197918896 \beta_{14} + 5004912292083 \beta_{15} - 3981667607415 \beta_{16} - 150818395905 \beta_{17} + 49313140848 \beta_{18} + 73558205352 \beta_{19}) q^{36} +(-\)\(83\!\cdots\!44\)\( - \)\(68\!\cdots\!98\)\( \beta_{1} - \)\(13\!\cdots\!17\)\( \beta_{2} + \)\(12\!\cdots\!90\)\( \beta_{3} - \)\(43\!\cdots\!57\)\( \beta_{4} - \)\(61\!\cdots\!51\)\( \beta_{5} - \)\(64\!\cdots\!69\)\( \beta_{6} + \)\(26\!\cdots\!03\)\( \beta_{7} - 1663470146959733902 \beta_{8} - 10860775574289919511 \beta_{9} + 597940116912932829 \beta_{10} - 3891710290422092 \beta_{11} + 3367897930955684 \beta_{12} - 3436530795273 \beta_{13} - 545323677111642 \beta_{14} + 8144910052398 \beta_{15} - 825741494052 \beta_{16} + 217669224348 \beta_{17} + 1216144539408 \beta_{18} + 304036134852 \beta_{19}) q^{37} +(-\)\(13\!\cdots\!37\)\( + \)\(14\!\cdots\!07\)\( \beta_{1} + \)\(33\!\cdots\!01\)\( \beta_{2} + \)\(17\!\cdots\!96\)\( \beta_{3} - \)\(89\!\cdots\!59\)\( \beta_{4} + \)\(18\!\cdots\!07\)\( \beta_{5} - \)\(14\!\cdots\!04\)\( \beta_{6} + \)\(38\!\cdots\!00\)\( \beta_{7} - \)\(15\!\cdots\!40\)\( \beta_{8} + 6794233374392039391 \beta_{9} + 284640329808496258 \beta_{10} - 73628635606209911 \beta_{11} + 13777152071802645 \beta_{12} + 9185804829600 \beta_{13} + 3631959857543406 \beta_{14} + 4557003513888 \beta_{15} - 758348859262 \beta_{16} + 1542933771904 \beta_{17} - 1222743489600 \beta_{18} + 611371744800 \beta_{19}) q^{38} +(\)\(22\!\cdots\!78\)\( - \)\(18\!\cdots\!57\)\( \beta_{1} - \)\(35\!\cdots\!65\)\( \beta_{2} + \)\(64\!\cdots\!04\)\( \beta_{3} - \)\(39\!\cdots\!32\)\( \beta_{4} + \)\(72\!\cdots\!90\)\( \beta_{5} + \)\(27\!\cdots\!06\)\( \beta_{6} + \)\(47\!\cdots\!49\)\( \beta_{7} + \)\(42\!\cdots\!72\)\( \beta_{8} + 18963036502202621327 \beta_{9} - 15315635630653143 \beta_{10} + 50211478228931804 \beta_{11} + 32477187650970633 \beta_{12} - 679198384546449 \beta_{13} - 451495517090827 \beta_{14} - 24079414575300 \beta_{15} + 43042783961223 \beta_{16} + 3717208198359 \beta_{17} - 524089797021 \beta_{18} - 764130717279 \beta_{19}) q^{39} +(-\)\(15\!\cdots\!54\)\( - \)\(21\!\cdots\!42\)\( \beta_{1} - \)\(43\!\cdots\!56\)\( \beta_{2} + \)\(23\!\cdots\!28\)\( \beta_{3} - \)\(12\!\cdots\!74\)\( \beta_{4} - \)\(13\!\cdots\!94\)\( \beta_{5} - \)\(96\!\cdots\!88\)\( \beta_{6} + \)\(63\!\cdots\!52\)\( \beta_{7} - 72111973118134081248 \beta_{8} + \)\(17\!\cdots\!12\)\( \beta_{9} + 86284049455686614 \beta_{10} - 103377981872896802 \beta_{11} + 111623335441532788 \beta_{12} + 632581674868510 \beta_{13} - 6327552557036256 \beta_{14} - 114810279772966 \beta_{15} + 4119634622884 \beta_{16} + 3310956128304 \beta_{17} - 14861181502376 \beta_{18} - 3715295375594 \beta_{19}) q^{40} +(\)\(34\!\cdots\!28\)\( + \)\(13\!\cdots\!87\)\( \beta_{1} - \)\(87\!\cdots\!38\)\( \beta_{2} - \)\(26\!\cdots\!24\)\( \beta_{3} + \)\(77\!\cdots\!96\)\( \beta_{4} + \)\(23\!\cdots\!68\)\( \beta_{5} - \)\(18\!\cdots\!68\)\( \beta_{6} + \)\(79\!\cdots\!79\)\( \beta_{7} - \)\(23\!\cdots\!79\)\( \beta_{8} - \)\(46\!\cdots\!54\)\( \beta_{9} + 5938125762725385952 \beta_{10} + 356160353359274912 \beta_{11} + 233703776281968675 \beta_{12} - 325454474816271 \beta_{13} - 22128931764161406 \beta_{14} - 300326100288516 \beta_{15} - 90269496073041 \beta_{16} - 8376124842585 \beta_{17} + 17883581038830 \beta_{18} - 8941790519415 \beta_{19}) q^{41} +(-\)\(11\!\cdots\!48\)\( - \)\(16\!\cdots\!21\)\( \beta_{1} - \)\(59\!\cdots\!95\)\( \beta_{2} + \)\(69\!\cdots\!51\)\( \beta_{3} - \)\(27\!\cdots\!16\)\( \beta_{4} - \)\(75\!\cdots\!80\)\( \beta_{5} - \)\(16\!\cdots\!98\)\( \beta_{6} - \)\(87\!\cdots\!95\)\( \beta_{7} + \)\(24\!\cdots\!24\)\( \beta_{8} + \)\(10\!\cdots\!31\)\( \beta_{9} + 50020868599908769965 \beta_{10} - 2032855687412605188 \beta_{11} + 593064577312560402 \beta_{12} + 5360313187474882 \beta_{13} - 283820640868504 \beta_{14} - 548288692451388 \beta_{15} - 342630277434558 \beta_{16} - 41198342665914 \beta_{17} + 4152266820968 \beta_{18} + 5978111296122 \beta_{19}) q^{42} +(\)\(51\!\cdots\!60\)\( + \)\(40\!\cdots\!02\)\( \beta_{1} + \)\(82\!\cdots\!42\)\( \beta_{2} + \)\(99\!\cdots\!71\)\( \beta_{3} + \)\(22\!\cdots\!85\)\( \beta_{4} - \)\(11\!\cdots\!64\)\( \beta_{5} + \)\(28\!\cdots\!56\)\( \beta_{6} - \)\(11\!\cdots\!06\)\( \beta_{7} - \)\(56\!\cdots\!33\)\( \beta_{8} + \)\(47\!\cdots\!26\)\( \beta_{9} - 27369002889775603684 \beta_{10} - 1026570803424413090 \beta_{11} + 1048182292904620287 \beta_{12} - 11964267197613739 \beta_{13} + 153491753461716072 \beta_{14} - 524959650109352 \beta_{15} + 4921116286583 \beta_{16} - 78633899253081 \beta_{17} + 147425565932996 \beta_{18} + 36856391483249 \beta_{19}) q^{43} +(\)\(76\!\cdots\!21\)\( - \)\(93\!\cdots\!98\)\( \beta_{1} - \)\(19\!\cdots\!82\)\( \beta_{2} - \)\(84\!\cdots\!72\)\( \beta_{3} + \)\(62\!\cdots\!45\)\( \beta_{4} + \)\(27\!\cdots\!39\)\( \beta_{5} - \)\(21\!\cdots\!96\)\( \beta_{6} - \)\(14\!\cdots\!85\)\( \beta_{7} + \)\(11\!\cdots\!01\)\( \beta_{8} + \)\(89\!\cdots\!90\)\( \beta_{9} + 39716640567456759031 \beta_{10} - 3680676240931509697 \beta_{11} + 2495380838550827561 \beta_{12} + 462545141663592 \beta_{13} - 71674644047549148 \beta_{14} + 590371823619207 \beta_{15} + 1844461076973397 \beta_{16} - 42608893985205 \beta_{17} - 213721976612160 \beta_{18} + 106860988306080 \beta_{19}) q^{44} +(\)\(18\!\cdots\!13\)\( + \)\(36\!\cdots\!36\)\( \beta_{1} + \)\(15\!\cdots\!00\)\( \beta_{2} + \)\(23\!\cdots\!12\)\( \beta_{3} - \)\(13\!\cdots\!31\)\( \beta_{4} - \)\(36\!\cdots\!30\)\( \beta_{5} - \)\(29\!\cdots\!30\)\( \beta_{6} + \)\(58\!\cdots\!99\)\( \beta_{7} - \)\(34\!\cdots\!19\)\( \beta_{8} - \)\(17\!\cdots\!06\)\( \beta_{9} - \)\(18\!\cdots\!64\)\( \beta_{10} + 4061135920815898684 \beta_{11} + 4393149328996840547 \beta_{12} - 27162597125627633 \beta_{13} + 9422754799927112 \beta_{14} + 4402925771877286 \beta_{15} + 1784595911624811 \beta_{16} + 228210212228931 \beta_{17} - 21613419230166 \beta_{18} - 31033133888211 \beta_{19}) q^{45} +(-\)\(14\!\cdots\!30\)\( - \)\(56\!\cdots\!90\)\( \beta_{1} - \)\(11\!\cdots\!86\)\( \beta_{2} + \)\(13\!\cdots\!20\)\( \beta_{3} - \)\(32\!\cdots\!62\)\( \beta_{4} - \)\(11\!\cdots\!02\)\( \beta_{5} + \)\(72\!\cdots\!00\)\( \beta_{6} + \)\(11\!\cdots\!56\)\( \beta_{7} - \)\(42\!\cdots\!64\)\( \beta_{8} - \)\(16\!\cdots\!46\)\( \beta_{9} + \)\(76\!\cdots\!00\)\( \beta_{10} - 9620193451018420638 \beta_{11} + 8969004507534812350 \beta_{12} + 116220835577107104 \beta_{13} - 1515079170721444032 \beta_{14} + 11764927985432128 \beta_{15} - 190165769138464 \beta_{16} + 779258878843104 \beta_{17} - 1178186219409280 \beta_{18} - 294546554852320 \beta_{19}) q^{46} +(-\)\(33\!\cdots\!38\)\( - \)\(31\!\cdots\!92\)\( \beta_{1} + \)\(82\!\cdots\!34\)\( \beta_{2} + \)\(48\!\cdots\!22\)\( \beta_{3} + \)\(16\!\cdots\!78\)\( \beta_{4} + \)\(12\!\cdots\!84\)\( \beta_{5} - \)\(97\!\cdots\!82\)\( \beta_{6} + \)\(12\!\cdots\!54\)\( \beta_{7} + \)\(14\!\cdots\!02\)\( \beta_{8} + \)\(25\!\cdots\!18\)\( \beta_{9} + 78885142815101843614 \beta_{10} - 35336133166085438158 \beta_{11} + 10311828457948844150 \beta_{12} + 24441583381705170 \beta_{13} + 2649593933276022468 \beta_{14} + 20761675647607224 \beta_{15} - 22412461803097826 \beta_{16} + 1226635911365982 \beta_{17} + 2131217182195740 \beta_{18} - 1065608591097870 \beta_{19}) q^{47} +(-\)\(63\!\cdots\!89\)\( + \)\(20\!\cdots\!01\)\( \beta_{1} + \)\(14\!\cdots\!70\)\( \beta_{2} + \)\(18\!\cdots\!30\)\( \beta_{3} - \)\(13\!\cdots\!91\)\( \beta_{4} - \)\(78\!\cdots\!09\)\( \beta_{5} - \)\(10\!\cdots\!98\)\( \beta_{6} + \)\(10\!\cdots\!52\)\( \beta_{7} - \)\(16\!\cdots\!00\)\( \beta_{8} - \)\(96\!\cdots\!82\)\( \beta_{9} - \)\(63\!\cdots\!71\)\( \beta_{10} + 5874432802705128249 \beta_{11} + 9485481449965646742 \beta_{12} + 29913188795360409 \beta_{13} + 110859168418097520 \beta_{14} + 18521722470486315 \beta_{15} - 1821085579507410 \beta_{16} + 119778308927784 \beta_{17} + 22512101051988 \beta_{18} + 34504832919357 \beta_{19}) q^{48} +(\)\(10\!\cdots\!09\)\( + \)\(25\!\cdots\!18\)\( \beta_{1} + \)\(51\!\cdots\!00\)\( \beta_{2} - \)\(10\!\cdots\!23\)\( \beta_{3} + \)\(14\!\cdots\!08\)\( \beta_{4} + \)\(19\!\cdots\!33\)\( \beta_{5} + \)\(77\!\cdots\!54\)\( \beta_{6} - \)\(61\!\cdots\!57\)\( \beta_{7} + \)\(78\!\cdots\!22\)\( \beta_{8} + \)\(14\!\cdots\!65\)\( \beta_{9} - \)\(23\!\cdots\!36\)\( \beta_{10} + 13447103486532951144 \beta_{11} - 12992183774004362606 \beta_{12} - 605748892353451077 \beta_{13} + 7940339132366526417 \beta_{14} - 16965834970556003 \beta_{15} + 374163910879388 \beta_{16} - 4030536734789796 \beta_{17} + 7312745647820816 \beta_{18} + 1828186411955204 \beta_{19}) q^{49} +(\)\(77\!\cdots\!80\)\( - \)\(79\!\cdots\!45\)\( \beta_{1} - \)\(19\!\cdots\!90\)\( \beta_{2} - \)\(77\!\cdots\!80\)\( \beta_{3} + \)\(93\!\cdots\!80\)\( \beta_{4} - \)\(66\!\cdots\!60\)\( \beta_{5} + \)\(51\!\cdots\!00\)\( \beta_{6} - \)\(17\!\cdots\!80\)\( \beta_{7} - \)\(47\!\cdots\!90\)\( \beta_{8} - \)\(14\!\cdots\!00\)\( \beta_{9} - \)\(41\!\cdots\!40\)\( \beta_{10} + \)\(26\!\cdots\!30\)\( \beta_{11} - 49687558962201277600 \beta_{12} - 156482532591293160 \beta_{13} - 24344911138805202510 \beta_{14} - 122470961568631560 \beta_{15} + 197168116415918590 \beta_{16} - 11337190340887200 \beta_{17} - 17948954312924400 \beta_{18} + 8974477156462200 \beta_{19}) q^{50} +(-\)\(44\!\cdots\!95\)\( - \)\(88\!\cdots\!43\)\( \beta_{1} - \)\(28\!\cdots\!01\)\( \beta_{2} + \)\(26\!\cdots\!89\)\( \beta_{3} - \)\(97\!\cdots\!01\)\( \beta_{4} + \)\(16\!\cdots\!22\)\( \beta_{5} + \)\(79\!\cdots\!79\)\( \beta_{6} - \)\(34\!\cdots\!90\)\( \beta_{7} + \)\(18\!\cdots\!70\)\( \beta_{8} + \)\(13\!\cdots\!03\)\( \beta_{9} + \)\(15\!\cdots\!77\)\( \beta_{10} - \)\(43\!\cdots\!27\)\( \beta_{11} - \)\(14\!\cdots\!48\)\( \beta_{12} + 1299323404187045900 \beta_{13} - 973445362615452650 \beta_{14} - 292412170697700456 \beta_{15} - 77062645454376816 \beta_{16} - 13434816504909456 \beta_{17} + 920587562118322 \beta_{18} + 1293191490710448 \beta_{19}) q^{51} +(\)\(62\!\cdots\!94\)\( + \)\(22\!\cdots\!60\)\( \beta_{1} + \)\(45\!\cdots\!52\)\( \beta_{2} - \)\(70\!\cdots\!94\)\( \beta_{3} + \)\(14\!\cdots\!60\)\( \beta_{4} + \)\(42\!\cdots\!40\)\( \beta_{5} - \)\(36\!\cdots\!96\)\( \beta_{6} + \)\(71\!\cdots\!36\)\( \beta_{7} + \)\(14\!\cdots\!92\)\( \beta_{8} + \)\(76\!\cdots\!16\)\( \beta_{9} - \)\(19\!\cdots\!32\)\( \beta_{10} + \)\(37\!\cdots\!04\)\( \beta_{11} - \)\(34\!\cdots\!72\)\( \beta_{12} + 764512417620864992 \beta_{13} - 4237681252089345624 \beta_{14} - 476481877265752532 \beta_{15} + 13267496407163588 \beta_{16} + 2214067613335740 \beta_{17} - 30963128040998656 \beta_{18} - 7740782010249664 \beta_{19}) q^{52} +(-\)\(55\!\cdots\!49\)\( + \)\(42\!\cdots\!05\)\( \beta_{1} + \)\(13\!\cdots\!42\)\( \beta_{2} + \)\(77\!\cdots\!56\)\( \beta_{3} + \)\(17\!\cdots\!09\)\( \beta_{4} - \)\(78\!\cdots\!88\)\( \beta_{5} + \)\(61\!\cdots\!04\)\( \beta_{6} + \)\(15\!\cdots\!42\)\( \beta_{7} - \)\(79\!\cdots\!14\)\( \beta_{8} - \)\(67\!\cdots\!16\)\( \beta_{9} - \)\(91\!\cdots\!08\)\( \beta_{10} + \)\(79\!\cdots\!36\)\( \beta_{11} - \)\(75\!\cdots\!70\)\( \beta_{12} - 230716084127734950 \beta_{13} + \)\(11\!\cdots\!44\)\( \beta_{14} - 396941942283371688 \beta_{15} - 1321993896272424538 \beta_{16} + 55408619385212246 \beta_{17} + 128408142057221100 \beta_{18} - 64204071028610550 \beta_{19}) q^{53} +(\)\(69\!\cdots\!47\)\( + \)\(16\!\cdots\!88\)\( \beta_{1} - \)\(19\!\cdots\!74\)\( \beta_{2} + \)\(13\!\cdots\!30\)\( \beta_{3} - \)\(68\!\cdots\!27\)\( \beta_{4} + \)\(20\!\cdots\!50\)\( \beta_{5} + \)\(99\!\cdots\!68\)\( \beta_{6} + \)\(19\!\cdots\!94\)\( \beta_{7} + \)\(50\!\cdots\!90\)\( \beta_{8} + \)\(17\!\cdots\!21\)\( \beta_{9} - \)\(92\!\cdots\!22\)\( \beta_{10} + \)\(51\!\cdots\!87\)\( \beta_{11} - \)\(14\!\cdots\!79\)\( \beta_{12} - 16729847069372086446 \beta_{13} - 5640610565670461694 \beta_{14} + 245108034256555044 \beta_{15} + 961632057523396104 \beta_{16} + 117330096478000950 \beta_{17} - 11510198049737304 \beta_{18} - 16442092331269686 \beta_{19}) q^{54} +(\)\(43\!\cdots\!56\)\( - \)\(62\!\cdots\!85\)\( \beta_{1} - \)\(12\!\cdots\!26\)\( \beta_{2} - \)\(82\!\cdots\!20\)\( \beta_{3} - \)\(36\!\cdots\!90\)\( \beta_{4} + \)\(19\!\cdots\!94\)\( \beta_{5} - \)\(28\!\cdots\!36\)\( \beta_{6} + \)\(18\!\cdots\!67\)\( \beta_{7} + \)\(10\!\cdots\!21\)\( \beta_{8} - \)\(24\!\cdots\!42\)\( \beta_{9} + \)\(12\!\cdots\!12\)\( \beta_{10} + \)\(17\!\cdots\!98\)\( \beta_{11} - \)\(18\!\cdots\!07\)\( \beta_{12} + 14243438208352368455 \beta_{13} - \)\(29\!\cdots\!16\)\( \beta_{14} + 2132492242605509604 \beta_{15} - 153312270064207371 \beta_{16} + 138677743266231429 \beta_{17} + 29269053595951884 \beta_{18} + 7317263398987971 \beta_{19}) q^{55} +(\)\(20\!\cdots\!58\)\( - \)\(40\!\cdots\!30\)\( \beta_{1} - \)\(51\!\cdots\!46\)\( \beta_{2} - \)\(21\!\cdots\!98\)\( \beta_{3} + \)\(22\!\cdots\!56\)\( \beta_{4} - \)\(20\!\cdots\!52\)\( \beta_{5} + \)\(15\!\cdots\!16\)\( \beta_{6} - \)\(46\!\cdots\!18\)\( \beta_{7} + \)\(35\!\cdots\!58\)\( \beta_{8} + \)\(84\!\cdots\!52\)\( \beta_{9} - \)\(71\!\cdots\!60\)\( \beta_{10} - \)\(90\!\cdots\!72\)\( \beta_{11} - \)\(20\!\cdots\!60\)\( \beta_{12} + 4821474172982403888 \beta_{13} - 63577990595402402256 \beta_{14} + 5037213575722566168 \beta_{15} + 6792009072765175464 \beta_{16} - 71913134246720760 \beta_{17} - 780398476371047520 \beta_{18} + 390199238185523760 \beta_{19}) q^{56} +(-\)\(20\!\cdots\!77\)\( + \)\(45\!\cdots\!32\)\( \beta_{1} + \)\(45\!\cdots\!64\)\( \beta_{2} + \)\(25\!\cdots\!25\)\( \beta_{3} - \)\(13\!\cdots\!94\)\( \beta_{4} + \)\(54\!\cdots\!08\)\( \beta_{5} - \)\(36\!\cdots\!59\)\( \beta_{6} - \)\(60\!\cdots\!96\)\( \beta_{7} - \)\(63\!\cdots\!01\)\( \beta_{8} - \)\(63\!\cdots\!61\)\( \beta_{9} - \)\(26\!\cdots\!20\)\( \beta_{10} - \)\(36\!\cdots\!12\)\( \beta_{11} - \)\(58\!\cdots\!03\)\( \beta_{12} + \)\(10\!\cdots\!09\)\( \beta_{13} + 97802868179162992716 \beta_{14} + 8786879482477261650 \beta_{15} - 6793167204230321655 \beta_{16} - 479084157335128815 \beta_{17} + 82065602700860202 \beta_{18} + 119551172398223103 \beta_{19}) q^{57} +(\)\(58\!\cdots\!04\)\( + \)\(78\!\cdots\!19\)\( \beta_{1} + \)\(15\!\cdots\!85\)\( \beta_{2} - \)\(63\!\cdots\!13\)\( \beta_{3} + \)\(43\!\cdots\!44\)\( \beta_{4} - \)\(48\!\cdots\!64\)\( \beta_{5} + \)\(90\!\cdots\!62\)\( \beta_{6} - \)\(21\!\cdots\!85\)\( \beta_{7} + \)\(95\!\cdots\!78\)\( \beta_{8} - \)\(17\!\cdots\!01\)\( \beta_{9} - \)\(37\!\cdots\!75\)\( \beta_{10} - \)\(20\!\cdots\!80\)\( \beta_{11} + \)\(12\!\cdots\!72\)\( \beta_{12} - \)\(14\!\cdots\!60\)\( \beta_{13} + \)\(24\!\cdots\!00\)\( \beta_{14} + 8557431351914340088 \beta_{15} + 698845448012910068 \beta_{16} - 1159891574229929484 \beta_{17} + 922092252434038832 \beta_{18} + 230523063108509708 \beta_{19}) q^{58} +(\)\(11\!\cdots\!50\)\( - \)\(55\!\cdots\!67\)\( \beta_{1} - \)\(27\!\cdots\!30\)\( \beta_{2} - \)\(15\!\cdots\!01\)\( \beta_{3} - \)\(41\!\cdots\!49\)\( \beta_{4} + \)\(83\!\cdots\!14\)\( \beta_{5} - \)\(64\!\cdots\!52\)\( \beta_{6} - \)\(32\!\cdots\!07\)\( \beta_{7} + \)\(16\!\cdots\!32\)\( \beta_{8} - \)\(64\!\cdots\!72\)\( \beta_{9} + \)\(13\!\cdots\!60\)\( \beta_{10} + \)\(70\!\cdots\!16\)\( \beta_{11} + \)\(11\!\cdots\!04\)\( \beta_{12} - 2589921434078498052 \beta_{13} - \)\(26\!\cdots\!44\)\( \beta_{14} + 364896630744565968 \beta_{15} - 25883753481349708444 \beta_{16} - 984939354941021340 \beta_{17} + 4002387898387693320 \beta_{18} - 2001193949193846660 \beta_{19}) q^{59} +(-\)\(22\!\cdots\!89\)\( + \)\(29\!\cdots\!90\)\( \beta_{1} - \)\(47\!\cdots\!30\)\( \beta_{2} + \)\(26\!\cdots\!08\)\( \beta_{3} + \)\(77\!\cdots\!43\)\( \beta_{4} - \)\(97\!\cdots\!43\)\( \beta_{5} - \)\(39\!\cdots\!20\)\( \beta_{6} - \)\(15\!\cdots\!39\)\( \beta_{7} - \)\(69\!\cdots\!81\)\( \beta_{8} + \)\(32\!\cdots\!62\)\( \beta_{9} + \)\(32\!\cdots\!01\)\( \beta_{10} - \)\(11\!\cdots\!55\)\( \beta_{11} + \)\(28\!\cdots\!31\)\( \beta_{12} - \)\(31\!\cdots\!96\)\( \beta_{13} - \)\(41\!\cdots\!60\)\( \beta_{14} - 29606401034127251847 \beta_{15} + 29964096025193142603 \beta_{16} + 29895312056295573 \beta_{17} - 374362792583697856 \beta_{18} - 565197404581718208 \beta_{19}) q^{60} +(\)\(22\!\cdots\!84\)\( - \)\(28\!\cdots\!74\)\( \beta_{1} - \)\(57\!\cdots\!21\)\( \beta_{2} - \)\(26\!\cdots\!14\)\( \beta_{3} - \)\(17\!\cdots\!77\)\( \beta_{4} - \)\(76\!\cdots\!39\)\( \beta_{5} + \)\(11\!\cdots\!83\)\( \beta_{6} + \)\(62\!\cdots\!95\)\( \beta_{7} - \)\(44\!\cdots\!82\)\( \beta_{8} + \)\(11\!\cdots\!45\)\( \beta_{9} - \)\(73\!\cdots\!71\)\( \beta_{10} - \)\(66\!\cdots\!72\)\( \beta_{11} + \)\(71\!\cdots\!76\)\( \beta_{12} + \)\(79\!\cdots\!39\)\( \beta_{13} - \)\(91\!\cdots\!34\)\( \beta_{14} - 75648034672387575974 \beta_{15} + 743268487529174804 \beta_{16} + 4496461054704999252 \beta_{17} - 10479459084468348112 \beta_{18} - 2619864771117087028 \beta_{19}) q^{61} +(-\)\(76\!\cdots\!73\)\( + \)\(90\!\cdots\!50\)\( \beta_{1} + \)\(19\!\cdots\!14\)\( \beta_{2} + \)\(91\!\cdots\!96\)\( \beta_{3} - \)\(35\!\cdots\!39\)\( \beta_{4} + \)\(13\!\cdots\!77\)\( \beta_{5} - \)\(10\!\cdots\!06\)\( \beta_{6} + \)\(20\!\cdots\!32\)\( \beta_{7} - \)\(10\!\cdots\!59\)\( \beta_{8} - \)\(18\!\cdots\!81\)\( \beta_{9} + \)\(31\!\cdots\!06\)\( \beta_{10} - \)\(32\!\cdots\!62\)\( \beta_{11} + \)\(10\!\cdots\!05\)\( \beta_{12} - \)\(11\!\cdots\!84\)\( \beta_{13} + \)\(17\!\cdots\!75\)\( \beta_{14} - \)\(13\!\cdots\!92\)\( \beta_{15} + 66104004744161029193 \beta_{16} + 7385994620373758336 \beta_{17} - 17030461642097449920 \beta_{18} + 8515230821048724960 \beta_{19}) q^{62} +(-\)\(91\!\cdots\!50\)\( - \)\(10\!\cdots\!80\)\( \beta_{1} - \)\(74\!\cdots\!66\)\( \beta_{2} + \)\(12\!\cdots\!66\)\( \beta_{3} + \)\(60\!\cdots\!34\)\( \beta_{4} - \)\(22\!\cdots\!71\)\( \beta_{5} + \)\(97\!\cdots\!69\)\( \beta_{6} + \)\(13\!\cdots\!76\)\( \beta_{7} + \)\(12\!\cdots\!30\)\( \beta_{8} - \)\(15\!\cdots\!32\)\( \beta_{9} - \)\(19\!\cdots\!32\)\( \beta_{10} + \)\(46\!\cdots\!90\)\( \beta_{11} + \)\(13\!\cdots\!56\)\( \beta_{12} - \)\(64\!\cdots\!72\)\( \beta_{13} - 90149075558781809562 \beta_{14} - \)\(13\!\cdots\!64\)\( \beta_{15} - 45816106986499865544 \beta_{16} + 11897859351708814680 \beta_{17} + 706083845344961706 \beta_{18} + 1227433005926019144 \beta_{19}) q^{63} +(\)\(15\!\cdots\!92\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(21\!\cdots\!52\)\( \beta_{2} - \)\(22\!\cdots\!84\)\( \beta_{3} + \)\(56\!\cdots\!00\)\( \beta_{4} - \)\(13\!\cdots\!40\)\( \beta_{5} - \)\(10\!\cdots\!28\)\( \beta_{6} - \)\(23\!\cdots\!60\)\( \beta_{7} - \)\(73\!\cdots\!92\)\( \beta_{8} + \)\(15\!\cdots\!56\)\( \beta_{9} + \)\(15\!\cdots\!44\)\( \beta_{10} - \)\(72\!\cdots\!88\)\( \beta_{11} + \)\(73\!\cdots\!16\)\( \beta_{12} - \)\(28\!\cdots\!80\)\( \beta_{13} + \)\(48\!\cdots\!04\)\( \beta_{14} - 35890666776971937128 \beta_{15} - 32587503059416136272 \beta_{16} - 3465997346518708608 \beta_{17} + 72107000811869689760 \beta_{18} + 18026750202967422440 \beta_{19}) q^{64} +(\)\(24\!\cdots\!56\)\( + \)\(26\!\cdots\!67\)\( \beta_{1} - \)\(61\!\cdots\!34\)\( \beta_{2} - \)\(38\!\cdots\!42\)\( \beta_{3} - \)\(21\!\cdots\!80\)\( \beta_{4} + \)\(43\!\cdots\!58\)\( \beta_{5} - \)\(32\!\cdots\!12\)\( \beta_{6} - \)\(83\!\cdots\!35\)\( \beta_{7} - \)\(15\!\cdots\!79\)\( \beta_{8} + \)\(47\!\cdots\!92\)\( \beta_{9} - \)\(16\!\cdots\!72\)\( \beta_{10} - \)\(49\!\cdots\!48\)\( \beta_{11} + \)\(41\!\cdots\!79\)\( \beta_{12} + \)\(30\!\cdots\!03\)\( \beta_{13} - \)\(44\!\cdots\!04\)\( \beta_{14} + \)\(35\!\cdots\!22\)\( \beta_{15} - 83931889825520259953 \beta_{16} - 16942967047478188473 \beta_{17} + 57895082511597300654 \beta_{18} - 28947541255798650327 \beta_{19}) q^{65} +(-\)\(36\!\cdots\!62\)\( - \)\(36\!\cdots\!87\)\( \beta_{1} - \)\(44\!\cdots\!50\)\( \beta_{2} + \)\(47\!\cdots\!43\)\( \beta_{3} + \)\(30\!\cdots\!12\)\( \beta_{4} + \)\(15\!\cdots\!00\)\( \beta_{5} + \)\(77\!\cdots\!32\)\( \beta_{6} - \)\(22\!\cdots\!74\)\( \beta_{7} + \)\(20\!\cdots\!64\)\( \beta_{8} - \)\(72\!\cdots\!85\)\( \beta_{9} - \)\(39\!\cdots\!21\)\( \beta_{10} - \)\(89\!\cdots\!25\)\( \beta_{11} - \)\(11\!\cdots\!07\)\( \beta_{12} + \)\(12\!\cdots\!61\)\( \beta_{13} + \)\(77\!\cdots\!49\)\( \beta_{14} + \)\(89\!\cdots\!22\)\( \beta_{15} - \)\(49\!\cdots\!64\)\( \beta_{16} - 73857358346979252429 \beta_{17} + 5122454564920848300 \beta_{18} + 6453861151971008385 \beta_{19}) q^{66} +(\)\(54\!\cdots\!72\)\( + \)\(11\!\cdots\!07\)\( \beta_{1} + \)\(23\!\cdots\!56\)\( \beta_{2} - \)\(59\!\cdots\!49\)\( \beta_{3} + \)\(66\!\cdots\!95\)\( \beta_{4} + \)\(29\!\cdots\!78\)\( \beta_{5} - \)\(85\!\cdots\!64\)\( \beta_{6} - \)\(26\!\cdots\!93\)\( \beta_{7} + \)\(41\!\cdots\!90\)\( \beta_{8} - \)\(27\!\cdots\!80\)\( \beta_{9} - \)\(28\!\cdots\!68\)\( \beta_{10} + \)\(27\!\cdots\!28\)\( \beta_{11} - \)\(39\!\cdots\!26\)\( \beta_{12} + \)\(36\!\cdots\!70\)\( \beta_{13} + \)\(15\!\cdots\!84\)\( \beta_{14} + \)\(16\!\cdots\!08\)\( \beta_{15} + \)\(23\!\cdots\!98\)\( \beta_{16} - 52711640765893984518 \beta_{17} - \)\(36\!\cdots\!60\)\( \beta_{18} - 91487461162654048490 \beta_{19}) q^{67} +(\)\(17\!\cdots\!44\)\( - \)\(13\!\cdots\!16\)\( \beta_{1} - \)\(43\!\cdots\!48\)\( \beta_{2} - \)\(22\!\cdots\!04\)\( \beta_{3} - \)\(34\!\cdots\!84\)\( \beta_{4} - \)\(19\!\cdots\!16\)\( \beta_{5} + \)\(15\!\cdots\!24\)\( \beta_{6} - \)\(51\!\cdots\!36\)\( \beta_{7} + \)\(16\!\cdots\!08\)\( \beta_{8} + \)\(23\!\cdots\!04\)\( \beta_{9} - \)\(46\!\cdots\!04\)\( \beta_{10} + \)\(70\!\cdots\!68\)\( \beta_{11} - \)\(15\!\cdots\!40\)\( \beta_{12} + \)\(14\!\cdots\!56\)\( \beta_{13} - \)\(14\!\cdots\!80\)\( \beta_{14} + \)\(17\!\cdots\!88\)\( \beta_{15} + \)\(12\!\cdots\!48\)\( \beta_{16} - 78327894023147854044 \beta_{17} - \)\(14\!\cdots\!40\)\( \beta_{18} + 71388186507010278720 \beta_{19}) q^{68} +(-\)\(12\!\cdots\!24\)\( + \)\(14\!\cdots\!55\)\( \beta_{1} - \)\(41\!\cdots\!58\)\( \beta_{2} + \)\(16\!\cdots\!50\)\( \beta_{3} + \)\(35\!\cdots\!94\)\( \beta_{4} - \)\(14\!\cdots\!08\)\( \beta_{5} - \)\(30\!\cdots\!94\)\( \beta_{6} - \)\(21\!\cdots\!55\)\( \beta_{7} - \)\(22\!\cdots\!47\)\( \beta_{8} + \)\(37\!\cdots\!40\)\( \beta_{9} + \)\(11\!\cdots\!28\)\( \beta_{10} - \)\(34\!\cdots\!36\)\( \beta_{11} - \)\(33\!\cdots\!73\)\( \beta_{12} - \)\(67\!\cdots\!31\)\( \beta_{13} - \)\(39\!\cdots\!82\)\( \beta_{14} + \)\(85\!\cdots\!56\)\( \beta_{15} + \)\(52\!\cdots\!27\)\( \beta_{16} + \)\(20\!\cdots\!87\)\( \beta_{17} - 61881245763052403590 \beta_{18} - 88742794702610624895 \beta_{19}) q^{69} +(\)\(27\!\cdots\!42\)\( + \)\(22\!\cdots\!80\)\( \beta_{1} + \)\(45\!\cdots\!98\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!50\)\( \beta_{4} + \)\(80\!\cdots\!98\)\( \beta_{5} + \)\(37\!\cdots\!28\)\( \beta_{6} - \)\(54\!\cdots\!66\)\( \beta_{7} + \)\(33\!\cdots\!42\)\( \beta_{8} + \)\(25\!\cdots\!86\)\( \beta_{9} - \)\(42\!\cdots\!96\)\( \beta_{10} + \)\(56\!\cdots\!86\)\( \beta_{11} - \)\(53\!\cdots\!04\)\( \beta_{12} + \)\(23\!\cdots\!90\)\( \beta_{13} - \)\(62\!\cdots\!52\)\( \beta_{14} - \)\(29\!\cdots\!92\)\( \beta_{15} - \)\(91\!\cdots\!42\)\( \beta_{16} + \)\(22\!\cdots\!78\)\( \beta_{17} + \)\(13\!\cdots\!28\)\( \beta_{18} + \)\(34\!\cdots\!82\)\( \beta_{19}) q^{70} +(\)\(77\!\cdots\!88\)\( - \)\(16\!\cdots\!23\)\( \beta_{1} - \)\(19\!\cdots\!76\)\( \beta_{2} - \)\(76\!\cdots\!04\)\( \beta_{3} + \)\(96\!\cdots\!54\)\( \beta_{4} - \)\(45\!\cdots\!24\)\( \beta_{5} + \)\(35\!\cdots\!38\)\( \beta_{6} - \)\(20\!\cdots\!07\)\( \beta_{7} + \)\(13\!\cdots\!93\)\( \beta_{8} - \)\(10\!\cdots\!12\)\( \beta_{9} - \)\(12\!\cdots\!34\)\( \beta_{10} - \)\(77\!\cdots\!90\)\( \beta_{11} - \)\(53\!\cdots\!49\)\( \beta_{12} - \)\(57\!\cdots\!11\)\( \beta_{13} + \)\(17\!\cdots\!98\)\( \beta_{14} - \)\(82\!\cdots\!36\)\( \beta_{15} - \)\(30\!\cdots\!17\)\( \beta_{16} + \)\(81\!\cdots\!75\)\( \beta_{17} + \)\(16\!\cdots\!50\)\( \beta_{18} - 83630331199872955575 \beta_{19}) q^{71} +(-\)\(28\!\cdots\!13\)\( + \)\(47\!\cdots\!89\)\( \beta_{1} + \)\(17\!\cdots\!21\)\( \beta_{2} + \)\(16\!\cdots\!83\)\( \beta_{3} - \)\(48\!\cdots\!38\)\( \beta_{4} - \)\(35\!\cdots\!40\)\( \beta_{5} - \)\(72\!\cdots\!80\)\( \beta_{6} - \)\(22\!\cdots\!57\)\( \beta_{7} - \)\(59\!\cdots\!83\)\( \beta_{8} - \)\(60\!\cdots\!28\)\( \beta_{9} + \)\(16\!\cdots\!34\)\( \beta_{10} + \)\(46\!\cdots\!30\)\( \beta_{11} - \)\(28\!\cdots\!16\)\( \beta_{12} + \)\(24\!\cdots\!82\)\( \beta_{13} + \)\(17\!\cdots\!04\)\( \beta_{14} - \)\(16\!\cdots\!06\)\( \beta_{15} - \)\(28\!\cdots\!56\)\( \beta_{16} - 53429667976469022624 \beta_{17} + \)\(36\!\cdots\!08\)\( \beta_{18} + \)\(55\!\cdots\!62\)\( \beta_{19}) q^{72} +(-\)\(28\!\cdots\!62\)\( + \)\(22\!\cdots\!44\)\( \beta_{1} + \)\(45\!\cdots\!78\)\( \beta_{2} + \)\(24\!\cdots\!22\)\( \beta_{3} + \)\(12\!\cdots\!22\)\( \beta_{4} - \)\(40\!\cdots\!04\)\( \beta_{5} + \)\(25\!\cdots\!90\)\( \beta_{6} - \)\(56\!\cdots\!56\)\( \beta_{7} - \)\(59\!\cdots\!16\)\( \beta_{8} + \)\(42\!\cdots\!32\)\( \beta_{9} + \)\(10\!\cdots\!42\)\( \beta_{10} - \)\(71\!\cdots\!68\)\( \beta_{11} + \)\(20\!\cdots\!68\)\( \beta_{12} - \)\(10\!\cdots\!56\)\( \beta_{13} - \)\(52\!\cdots\!26\)\( \beta_{14} - \)\(18\!\cdots\!70\)\( \beta_{15} + \)\(14\!\cdots\!40\)\( \beta_{16} + \)\(36\!\cdots\!96\)\( \beta_{17} - \)\(36\!\cdots\!72\)\( \beta_{18} - \)\(91\!\cdots\!68\)\( \beta_{19}) q^{73} +(\)\(64\!\cdots\!16\)\( - \)\(10\!\cdots\!08\)\( \beta_{1} - \)\(16\!\cdots\!30\)\( \beta_{2} - \)\(80\!\cdots\!60\)\( \beta_{3} + \)\(17\!\cdots\!04\)\( \beta_{4} + \)\(89\!\cdots\!40\)\( \beta_{5} - \)\(66\!\cdots\!60\)\( \beta_{6} - \)\(17\!\cdots\!60\)\( \beta_{7} - \)\(13\!\cdots\!74\)\( \beta_{8} - \)\(15\!\cdots\!52\)\( \beta_{9} + \)\(23\!\cdots\!84\)\( \beta_{10} - \)\(46\!\cdots\!42\)\( \beta_{11} + \)\(78\!\cdots\!52\)\( \beta_{12} - \)\(23\!\cdots\!32\)\( \beta_{13} - \)\(75\!\cdots\!74\)\( \beta_{14} - \)\(13\!\cdots\!32\)\( \beta_{15} + \)\(30\!\cdots\!46\)\( \beta_{16} - \)\(32\!\cdots\!00\)\( \beta_{17} + \)\(47\!\cdots\!00\)\( \beta_{18} - \)\(23\!\cdots\!00\)\( \beta_{19}) q^{74} +(\)\(11\!\cdots\!80\)\( - \)\(26\!\cdots\!40\)\( \beta_{1} - \)\(65\!\cdots\!50\)\( \beta_{2} - \)\(81\!\cdots\!85\)\( \beta_{3} - \)\(12\!\cdots\!65\)\( \beta_{4} - \)\(28\!\cdots\!40\)\( \beta_{5} - \)\(14\!\cdots\!95\)\( \beta_{6} - \)\(26\!\cdots\!45\)\( \beta_{7} + \)\(57\!\cdots\!15\)\( \beta_{8} + \)\(33\!\cdots\!85\)\( \beta_{9} - \)\(29\!\cdots\!15\)\( \beta_{10} - \)\(76\!\cdots\!55\)\( \beta_{11} + \)\(23\!\cdots\!35\)\( \beta_{12} - \)\(58\!\cdots\!95\)\( \beta_{13} - \)\(87\!\cdots\!40\)\( \beta_{14} + \)\(15\!\cdots\!60\)\( \beta_{15} + \)\(10\!\cdots\!35\)\( \beta_{16} - \)\(12\!\cdots\!45\)\( \beta_{17} - \)\(14\!\cdots\!20\)\( \beta_{18} - \)\(23\!\cdots\!55\)\( \beta_{19}) q^{75} +(-\)\(27\!\cdots\!25\)\( + \)\(18\!\cdots\!94\)\( \beta_{1} + \)\(37\!\cdots\!14\)\( \beta_{2} + \)\(30\!\cdots\!76\)\( \beta_{3} + \)\(10\!\cdots\!29\)\( \beta_{4} - \)\(72\!\cdots\!19\)\( \beta_{5} - \)\(43\!\cdots\!50\)\( \beta_{6} - \)\(46\!\cdots\!93\)\( \beta_{7} - \)\(24\!\cdots\!77\)\( \beta_{8} - \)\(10\!\cdots\!44\)\( \beta_{9} + \)\(21\!\cdots\!21\)\( \beta_{10} - \)\(57\!\cdots\!95\)\( \beta_{11} + \)\(52\!\cdots\!81\)\( \beta_{12} - \)\(29\!\cdots\!72\)\( \beta_{13} + \)\(15\!\cdots\!66\)\( \beta_{14} + \)\(59\!\cdots\!99\)\( \beta_{15} + \)\(54\!\cdots\!61\)\( \beta_{16} - \)\(71\!\cdots\!09\)\( \beta_{17} + \)\(32\!\cdots\!96\)\( \beta_{18} + \)\(82\!\cdots\!24\)\( \beta_{19}) q^{76} +(\)\(13\!\cdots\!34\)\( + \)\(63\!\cdots\!44\)\( \beta_{1} - \)\(32\!\cdots\!52\)\( \beta_{2} - \)\(17\!\cdots\!96\)\( \beta_{3} + \)\(97\!\cdots\!06\)\( \beta_{4} + \)\(10\!\cdots\!68\)\( \beta_{5} - \)\(78\!\cdots\!04\)\( \beta_{6} - \)\(36\!\cdots\!02\)\( \beta_{7} - \)\(70\!\cdots\!86\)\( \beta_{8} + \)\(13\!\cdots\!76\)\( \beta_{9} + \)\(79\!\cdots\!88\)\( \beta_{10} + \)\(19\!\cdots\!24\)\( \beta_{11} + \)\(10\!\cdots\!30\)\( \beta_{12} + \)\(14\!\cdots\!50\)\( \beta_{13} + \)\(15\!\cdots\!56\)\( \beta_{14} + \)\(12\!\cdots\!88\)\( \beta_{15} - \)\(17\!\cdots\!62\)\( \beta_{16} + \)\(59\!\cdots\!54\)\( \beta_{17} - \)\(32\!\cdots\!40\)\( \beta_{18} + \)\(16\!\cdots\!70\)\( \beta_{19}) q^{77} +(\)\(48\!\cdots\!40\)\( + \)\(11\!\cdots\!98\)\( \beta_{1} + \)\(88\!\cdots\!16\)\( \beta_{2} - \)\(81\!\cdots\!76\)\( \beta_{3} - \)\(90\!\cdots\!24\)\( \beta_{4} + \)\(27\!\cdots\!34\)\( \beta_{5} + \)\(10\!\cdots\!20\)\( \beta_{6} - \)\(93\!\cdots\!70\)\( \beta_{7} - \)\(11\!\cdots\!80\)\( \beta_{8} - \)\(51\!\cdots\!16\)\( \beta_{9} + \)\(91\!\cdots\!16\)\( \beta_{10} - \)\(57\!\cdots\!18\)\( \beta_{11} + \)\(13\!\cdots\!22\)\( \beta_{12} + \)\(94\!\cdots\!86\)\( \beta_{13} + \)\(27\!\cdots\!62\)\( \beta_{14} + \)\(16\!\cdots\!68\)\( \beta_{15} - \)\(21\!\cdots\!92\)\( \beta_{16} - \)\(19\!\cdots\!22\)\( \beta_{17} + \)\(35\!\cdots\!12\)\( \beta_{18} + \)\(61\!\cdots\!38\)\( \beta_{19}) q^{78} +(-\)\(11\!\cdots\!88\)\( + \)\(37\!\cdots\!43\)\( \beta_{1} + \)\(74\!\cdots\!04\)\( \beta_{2} + \)\(11\!\cdots\!16\)\( \beta_{3} + \)\(21\!\cdots\!86\)\( \beta_{4} - \)\(11\!\cdots\!83\)\( \beta_{5} - \)\(28\!\cdots\!59\)\( \beta_{6} - \)\(92\!\cdots\!07\)\( \beta_{7} - \)\(12\!\cdots\!49\)\( \beta_{8} - \)\(41\!\cdots\!94\)\( \beta_{9} - \)\(92\!\cdots\!28\)\( \beta_{10} - \)\(14\!\cdots\!42\)\( \beta_{11} + \)\(13\!\cdots\!35\)\( \beta_{12} + \)\(48\!\cdots\!09\)\( \beta_{13} - \)\(74\!\cdots\!84\)\( \beta_{14} + \)\(16\!\cdots\!72\)\( \beta_{15} - \)\(46\!\cdots\!33\)\( \beta_{16} + \)\(33\!\cdots\!23\)\( \beta_{17} + \)\(25\!\cdots\!20\)\( \beta_{18} + \)\(63\!\cdots\!05\)\( \beta_{19}) q^{79} +(\)\(99\!\cdots\!72\)\( - \)\(22\!\cdots\!76\)\( \beta_{1} - \)\(24\!\cdots\!28\)\( \beta_{2} - \)\(13\!\cdots\!04\)\( \beta_{3} - \)\(36\!\cdots\!60\)\( \beta_{4} + \)\(71\!\cdots\!56\)\( \beta_{5} - \)\(51\!\cdots\!04\)\( \beta_{6} - \)\(28\!\cdots\!60\)\( \beta_{7} + \)\(36\!\cdots\!12\)\( \beta_{8} + \)\(11\!\cdots\!64\)\( \beta_{9} + \)\(58\!\cdots\!56\)\( \beta_{10} - \)\(37\!\cdots\!76\)\( \beta_{11} + \)\(52\!\cdots\!68\)\( \beta_{12} - \)\(13\!\cdots\!04\)\( \beta_{13} - \)\(98\!\cdots\!48\)\( \beta_{14} - \)\(94\!\cdots\!56\)\( \beta_{15} + \)\(63\!\cdots\!44\)\( \beta_{16} - \)\(12\!\cdots\!16\)\( \beta_{17} + \)\(85\!\cdots\!68\)\( \beta_{18} - \)\(42\!\cdots\!84\)\( \beta_{19}) q^{80} +(\)\(18\!\cdots\!57\)\( + \)\(27\!\cdots\!72\)\( \beta_{1} + \)\(26\!\cdots\!14\)\( \beta_{2} - \)\(28\!\cdots\!39\)\( \beta_{3} + \)\(10\!\cdots\!86\)\( \beta_{4} - \)\(75\!\cdots\!87\)\( \beta_{5} + \)\(17\!\cdots\!54\)\( \beta_{6} - \)\(25\!\cdots\!51\)\( \beta_{7} - \)\(58\!\cdots\!86\)\( \beta_{8} + \)\(22\!\cdots\!79\)\( \beta_{9} + \)\(13\!\cdots\!62\)\( \beta_{10} + \)\(24\!\cdots\!96\)\( \beta_{11} - \)\(54\!\cdots\!98\)\( \beta_{12} - \)\(13\!\cdots\!09\)\( \beta_{13} - \)\(68\!\cdots\!53\)\( \beta_{14} - \)\(31\!\cdots\!29\)\( \beta_{15} - \)\(22\!\cdots\!52\)\( \beta_{16} + \)\(52\!\cdots\!40\)\( \beta_{17} - \)\(12\!\cdots\!28\)\( \beta_{18} - \)\(42\!\cdots\!92\)\( \beta_{19}) q^{81} +(-\)\(34\!\cdots\!12\)\( + \)\(30\!\cdots\!54\)\( \beta_{1} + \)\(61\!\cdots\!62\)\( \beta_{2} + \)\(48\!\cdots\!74\)\( \beta_{3} + \)\(17\!\cdots\!08\)\( \beta_{4} + \)\(29\!\cdots\!04\)\( \beta_{5} - \)\(63\!\cdots\!84\)\( \beta_{6} - \)\(73\!\cdots\!66\)\( \beta_{7} + \)\(82\!\cdots\!12\)\( \beta_{8} + \)\(17\!\cdots\!66\)\( \beta_{9} - \)\(15\!\cdots\!94\)\( \beta_{10} + \)\(35\!\cdots\!32\)\( \beta_{11} - \)\(30\!\cdots\!24\)\( \beta_{12} - \)\(23\!\cdots\!52\)\( \beta_{13} + \)\(18\!\cdots\!12\)\( \beta_{14} - \)\(88\!\cdots\!48\)\( \beta_{15} + \)\(16\!\cdots\!52\)\( \beta_{16} - \)\(82\!\cdots\!68\)\( \beta_{17} - \)\(16\!\cdots\!68\)\( \beta_{18} - \)\(41\!\cdots\!92\)\( \beta_{19}) q^{82} +(\)\(29\!\cdots\!27\)\( - \)\(23\!\cdots\!50\)\( \beta_{1} - \)\(73\!\cdots\!13\)\( \beta_{2} - \)\(39\!\cdots\!50\)\( \beta_{3} - \)\(41\!\cdots\!20\)\( \beta_{4} - \)\(58\!\cdots\!56\)\( \beta_{5} + \)\(46\!\cdots\!71\)\( \beta_{6} - \)\(86\!\cdots\!17\)\( \beta_{7} + \)\(20\!\cdots\!06\)\( \beta_{8} - \)\(19\!\cdots\!19\)\( \beta_{9} + \)\(20\!\cdots\!51\)\( \beta_{10} - \)\(26\!\cdots\!87\)\( \beta_{11} - \)\(72\!\cdots\!30\)\( \beta_{12} - \)\(12\!\cdots\!98\)\( \beta_{13} + \)\(35\!\cdots\!08\)\( \beta_{14} - \)\(13\!\cdots\!40\)\( \beta_{15} - \)\(13\!\cdots\!70\)\( \beta_{16} + \)\(13\!\cdots\!14\)\( \beta_{17} - \)\(81\!\cdots\!40\)\( \beta_{18} + \)\(40\!\cdots\!70\)\( \beta_{19}) q^{83} +(\)\(31\!\cdots\!01\)\( - \)\(14\!\cdots\!54\)\( \beta_{1} - \)\(17\!\cdots\!10\)\( \beta_{2} - \)\(45\!\cdots\!46\)\( \beta_{3} + \)\(91\!\cdots\!99\)\( \beta_{4} - \)\(37\!\cdots\!59\)\( \beta_{5} - \)\(85\!\cdots\!00\)\( \beta_{6} - \)\(20\!\cdots\!71\)\( \beta_{7} + \)\(61\!\cdots\!79\)\( \beta_{8} + \)\(10\!\cdots\!62\)\( \beta_{9} - \)\(83\!\cdots\!87\)\( \beta_{10} - \)\(32\!\cdots\!23\)\( \beta_{11} - \)\(14\!\cdots\!41\)\( \beta_{12} + \)\(95\!\cdots\!28\)\( \beta_{13} - \)\(31\!\cdots\!80\)\( \beta_{14} - \)\(16\!\cdots\!47\)\( \beta_{15} + \)\(20\!\cdots\!03\)\( \beta_{16} - \)\(27\!\cdots\!35\)\( \beta_{17} - \)\(33\!\cdots\!84\)\( \beta_{18} - \)\(53\!\cdots\!36\)\( \beta_{19}) q^{84} +(-\)\(71\!\cdots\!12\)\( + \)\(81\!\cdots\!36\)\( \beta_{1} + \)\(16\!\cdots\!92\)\( \beta_{2} - \)\(13\!\cdots\!24\)\( \beta_{3} + \)\(47\!\cdots\!92\)\( \beta_{4} + \)\(25\!\cdots\!96\)\( \beta_{5} + \)\(22\!\cdots\!48\)\( \beta_{6} - \)\(20\!\cdots\!64\)\( \beta_{7} + \)\(78\!\cdots\!80\)\( \beta_{8} + \)\(19\!\cdots\!12\)\( \beta_{9} + \)\(17\!\cdots\!20\)\( \beta_{10} + \)\(22\!\cdots\!64\)\( \beta_{11} - \)\(22\!\cdots\!56\)\( \beta_{12} + \)\(61\!\cdots\!20\)\( \beta_{13} - \)\(23\!\cdots\!28\)\( \beta_{14} - \)\(70\!\cdots\!48\)\( \beta_{15} - \)\(34\!\cdots\!48\)\( \beta_{16} + \)\(81\!\cdots\!72\)\( \beta_{17} + \)\(52\!\cdots\!52\)\( \beta_{18} + \)\(13\!\cdots\!88\)\( \beta_{19}) q^{85} +(\)\(12\!\cdots\!91\)\( + \)\(35\!\cdots\!23\)\( \beta_{1} - \)\(30\!\cdots\!19\)\( \beta_{2} - \)\(16\!\cdots\!20\)\( \beta_{3} - \)\(31\!\cdots\!59\)\( \beta_{4} - \)\(35\!\cdots\!45\)\( \beta_{5} + \)\(28\!\cdots\!44\)\( \beta_{6} - \)\(35\!\cdots\!76\)\( \beta_{7} - \)\(24\!\cdots\!96\)\( \beta_{8} + \)\(74\!\cdots\!51\)\( \beta_{9} - \)\(68\!\cdots\!90\)\( \beta_{10} + \)\(19\!\cdots\!21\)\( \beta_{11} - \)\(24\!\cdots\!07\)\( \beta_{12} - \)\(10\!\cdots\!00\)\( \beta_{13} - \)\(65\!\cdots\!10\)\( \beta_{14} + \)\(80\!\cdots\!00\)\( \beta_{15} - \)\(48\!\cdots\!10\)\( \beta_{16} - \)\(30\!\cdots\!00\)\( \beta_{17} - \)\(11\!\cdots\!00\)\( \beta_{18} + \)\(56\!\cdots\!00\)\( \beta_{19}) q^{86} +(\)\(11\!\cdots\!62\)\( - \)\(19\!\cdots\!46\)\( \beta_{1} + \)\(21\!\cdots\!47\)\( \beta_{2} - \)\(20\!\cdots\!60\)\( \beta_{3} - \)\(24\!\cdots\!66\)\( \beta_{4} + \)\(22\!\cdots\!45\)\( \beta_{5} - \)\(12\!\cdots\!23\)\( \beta_{6} - \)\(29\!\cdots\!46\)\( \beta_{7} + \)\(27\!\cdots\!83\)\( \beta_{8} - \)\(24\!\cdots\!87\)\( \beta_{9} - \)\(24\!\cdots\!59\)\( \beta_{10} + \)\(52\!\cdots\!90\)\( \beta_{11} - \)\(23\!\cdots\!68\)\( \beta_{12} - \)\(62\!\cdots\!48\)\( \beta_{13} + \)\(11\!\cdots\!81\)\( \beta_{14} + \)\(51\!\cdots\!96\)\( \beta_{15} - \)\(82\!\cdots\!14\)\( \beta_{16} + \)\(69\!\cdots\!06\)\( \beta_{17} + \)\(16\!\cdots\!75\)\( \beta_{18} + \)\(31\!\cdots\!70\)\( \beta_{19}) q^{87} +(\)\(17\!\cdots\!74\)\( + \)\(25\!\cdots\!30\)\( \beta_{1} + \)\(50\!\cdots\!72\)\( \beta_{2} - \)\(19\!\cdots\!96\)\( \beta_{3} + \)\(14\!\cdots\!18\)\( \beta_{4} + \)\(21\!\cdots\!26\)\( \beta_{5} + \)\(36\!\cdots\!40\)\( \beta_{6} - \)\(62\!\cdots\!44\)\( \beta_{7} + \)\(22\!\cdots\!28\)\( \beta_{8} - \)\(16\!\cdots\!96\)\( \beta_{9} - \)\(51\!\cdots\!38\)\( \beta_{10} - \)\(53\!\cdots\!10\)\( \beta_{11} - \)\(46\!\cdots\!80\)\( \beta_{12} - \)\(48\!\cdots\!18\)\( \beta_{13} + \)\(17\!\cdots\!04\)\( \beta_{14} + \)\(79\!\cdots\!70\)\( \beta_{15} + \)\(43\!\cdots\!80\)\( \beta_{16} - \)\(26\!\cdots\!04\)\( \beta_{17} - \)\(82\!\cdots\!52\)\( \beta_{18} - \)\(20\!\cdots\!38\)\( \beta_{19}) q^{88} +(\)\(21\!\cdots\!80\)\( + \)\(55\!\cdots\!24\)\( \beta_{1} - \)\(52\!\cdots\!50\)\( \beta_{2} - \)\(27\!\cdots\!57\)\( \beta_{3} + \)\(24\!\cdots\!66\)\( \beta_{4} + \)\(37\!\cdots\!11\)\( \beta_{5} - \)\(28\!\cdots\!06\)\( \beta_{6} - \)\(60\!\cdots\!53\)\( \beta_{7} - \)\(34\!\cdots\!90\)\( \beta_{8} + \)\(24\!\cdots\!97\)\( \beta_{9} - \)\(51\!\cdots\!58\)\( \beta_{10} - \)\(37\!\cdots\!76\)\( \beta_{11} + \)\(15\!\cdots\!02\)\( \beta_{12} + \)\(18\!\cdots\!17\)\( \beta_{13} + \)\(35\!\cdots\!13\)\( \beta_{14} + \)\(13\!\cdots\!57\)\( \beta_{15} + \)\(15\!\cdots\!68\)\( \beta_{16} + \)\(16\!\cdots\!20\)\( \beta_{17} - \)\(45\!\cdots\!60\)\( \beta_{18} + \)\(22\!\cdots\!80\)\( \beta_{19}) q^{89} +(-\)\(96\!\cdots\!52\)\( - \)\(21\!\cdots\!11\)\( \beta_{1} - \)\(88\!\cdots\!83\)\( \beta_{2} + \)\(92\!\cdots\!09\)\( \beta_{3} - \)\(32\!\cdots\!72\)\( \beta_{4} - \)\(76\!\cdots\!92\)\( \beta_{5} + \)\(55\!\cdots\!06\)\( \beta_{6} - \)\(14\!\cdots\!09\)\( \beta_{7} + \)\(44\!\cdots\!66\)\( \beta_{8} + \)\(10\!\cdots\!61\)\( \beta_{9} + \)\(16\!\cdots\!07\)\( \beta_{10} - \)\(28\!\cdots\!26\)\( \beta_{11} + \)\(83\!\cdots\!34\)\( \beta_{12} + \)\(19\!\cdots\!50\)\( \beta_{13} - \)\(12\!\cdots\!18\)\( \beta_{14} + \)\(80\!\cdots\!72\)\( \beta_{15} + \)\(15\!\cdots\!72\)\( \beta_{16} - \)\(65\!\cdots\!38\)\( \beta_{17} - \)\(40\!\cdots\!88\)\( \beta_{18} - \)\(92\!\cdots\!22\)\( \beta_{19}) q^{90} +(\)\(61\!\cdots\!40\)\( + \)\(53\!\cdots\!97\)\( \beta_{1} + \)\(10\!\cdots\!06\)\( \beta_{2} - \)\(83\!\cdots\!18\)\( \beta_{3} + \)\(30\!\cdots\!52\)\( \beta_{4} - \)\(18\!\cdots\!78\)\( \beta_{5} - \)\(12\!\cdots\!92\)\( \beta_{6} - \)\(12\!\cdots\!95\)\( \beta_{7} - \)\(71\!\cdots\!99\)\( \beta_{8} - \)\(86\!\cdots\!18\)\( \beta_{9} + \)\(36\!\cdots\!04\)\( \beta_{10} - \)\(15\!\cdots\!30\)\( \beta_{11} + \)\(15\!\cdots\!45\)\( \beta_{12} - \)\(27\!\cdots\!37\)\( \beta_{13} - \)\(21\!\cdots\!84\)\( \beta_{14} + \)\(63\!\cdots\!80\)\( \beta_{15} - \)\(51\!\cdots\!91\)\( \beta_{16} + \)\(10\!\cdots\!57\)\( \beta_{17} - \)\(10\!\cdots\!32\)\( \beta_{18} - \)\(27\!\cdots\!33\)\( \beta_{19}) q^{91} +(\)\(93\!\cdots\!90\)\( - \)\(26\!\cdots\!84\)\( \beta_{1} - \)\(23\!\cdots\!12\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!58\)\( \beta_{4} + \)\(28\!\cdots\!34\)\( \beta_{5} - \)\(21\!\cdots\!36\)\( \beta_{6} - \)\(25\!\cdots\!30\)\( \beta_{7} + \)\(19\!\cdots\!82\)\( \beta_{8} - \)\(55\!\cdots\!36\)\( \beta_{9} + \)\(71\!\cdots\!90\)\( \beta_{10} + \)\(41\!\cdots\!10\)\( \beta_{11} + \)\(27\!\cdots\!10\)\( \beta_{12} - \)\(38\!\cdots\!08\)\( \beta_{13} - \)\(85\!\cdots\!04\)\( \beta_{14} - \)\(26\!\cdots\!46\)\( \beta_{15} - \)\(64\!\cdots\!26\)\( \beta_{16} - \)\(40\!\cdots\!54\)\( \beta_{17} + \)\(72\!\cdots\!40\)\( \beta_{18} - \)\(36\!\cdots\!20\)\( \beta_{19}) q^{92} +(-\)\(13\!\cdots\!55\)\( + \)\(26\!\cdots\!07\)\( \beta_{1} + \)\(17\!\cdots\!89\)\( \beta_{2} + \)\(19\!\cdots\!70\)\( \beta_{3} - \)\(14\!\cdots\!74\)\( \beta_{4} + \)\(38\!\cdots\!53\)\( \beta_{5} + \)\(27\!\cdots\!51\)\( \beta_{6} - \)\(20\!\cdots\!17\)\( \beta_{7} - \)\(44\!\cdots\!22\)\( \beta_{8} - \)\(23\!\cdots\!51\)\( \beta_{9} - \)\(23\!\cdots\!83\)\( \beta_{10} + \)\(15\!\cdots\!16\)\( \beta_{11} + \)\(33\!\cdots\!32\)\( \beta_{12} - \)\(15\!\cdots\!57\)\( \beta_{13} - \)\(13\!\cdots\!94\)\( \beta_{14} - \)\(33\!\cdots\!46\)\( \beta_{15} - \)\(25\!\cdots\!76\)\( \beta_{16} - \)\(11\!\cdots\!32\)\( \beta_{17} + \)\(19\!\cdots\!04\)\( \beta_{18} + \)\(10\!\cdots\!36\)\( \beta_{19}) q^{93} +(\)\(82\!\cdots\!08\)\( + \)\(15\!\cdots\!60\)\( \beta_{1} + \)\(32\!\cdots\!20\)\( \beta_{2} - \)\(24\!\cdots\!64\)\( \beta_{3} + \)\(90\!\cdots\!24\)\( \beta_{4} - \)\(47\!\cdots\!44\)\( \beta_{5} - \)\(38\!\cdots\!84\)\( \beta_{6} - \)\(38\!\cdots\!12\)\( \beta_{7} - \)\(32\!\cdots\!68\)\( \beta_{8} + \)\(13\!\cdots\!96\)\( \beta_{9} + \)\(15\!\cdots\!36\)\( \beta_{10} - \)\(27\!\cdots\!60\)\( \beta_{11} + \)\(33\!\cdots\!60\)\( \beta_{12} + \)\(12\!\cdots\!80\)\( \beta_{13} + \)\(12\!\cdots\!36\)\( \beta_{14} - \)\(95\!\cdots\!80\)\( \beta_{15} + \)\(12\!\cdots\!88\)\( \beta_{16} - \)\(66\!\cdots\!60\)\( \beta_{17} + \)\(10\!\cdots\!44\)\( \beta_{18} + \)\(27\!\cdots\!36\)\( \beta_{19}) q^{94} +(\)\(10\!\cdots\!22\)\( - \)\(31\!\cdots\!13\)\( \beta_{1} - \)\(25\!\cdots\!86\)\( \beta_{2} - \)\(12\!\cdots\!86\)\( \beta_{3} + \)\(23\!\cdots\!48\)\( \beta_{4} + \)\(35\!\cdots\!44\)\( \beta_{5} - \)\(27\!\cdots\!76\)\( \beta_{6} - \)\(27\!\cdots\!75\)\( \beta_{7} + \)\(48\!\cdots\!73\)\( \beta_{8} - \)\(46\!\cdots\!34\)\( \beta_{9} + \)\(86\!\cdots\!84\)\( \beta_{10} - \)\(16\!\cdots\!84\)\( \beta_{11} + \)\(25\!\cdots\!67\)\( \beta_{12} - \)\(46\!\cdots\!71\)\( \beta_{13} + \)\(31\!\cdots\!18\)\( \beta_{14} - \)\(52\!\cdots\!84\)\( \beta_{15} + \)\(12\!\cdots\!91\)\( \beta_{16} + \)\(18\!\cdots\!71\)\( \beta_{17} - \)\(28\!\cdots\!58\)\( \beta_{18} + \)\(14\!\cdots\!29\)\( \beta_{19}) q^{95} +(-\)\(18\!\cdots\!08\)\( - \)\(28\!\cdots\!68\)\( \beta_{1} - \)\(10\!\cdots\!16\)\( \beta_{2} + \)\(26\!\cdots\!16\)\( \beta_{3} - \)\(69\!\cdots\!56\)\( \beta_{4} + \)\(53\!\cdots\!36\)\( \beta_{5} - \)\(19\!\cdots\!72\)\( \beta_{6} - \)\(16\!\cdots\!08\)\( \beta_{7} - \)\(57\!\cdots\!80\)\( \beta_{8} + \)\(37\!\cdots\!96\)\( \beta_{9} + \)\(10\!\cdots\!68\)\( \beta_{10} - \)\(68\!\cdots\!32\)\( \beta_{11} - \)\(46\!\cdots\!96\)\( \beta_{12} - \)\(21\!\cdots\!44\)\( \beta_{13} + \)\(57\!\cdots\!24\)\( \beta_{14} - \)\(11\!\cdots\!40\)\( \beta_{15} - \)\(59\!\cdots\!40\)\( \beta_{16} + \)\(16\!\cdots\!44\)\( \beta_{17} + \)\(23\!\cdots\!76\)\( \beta_{18} + \)\(39\!\cdots\!84\)\( \beta_{19}) q^{96} +(\)\(40\!\cdots\!24\)\( - \)\(53\!\cdots\!70\)\( \beta_{1} - \)\(10\!\cdots\!70\)\( \beta_{2} - \)\(56\!\cdots\!41\)\( \beta_{3} - \)\(30\!\cdots\!34\)\( \beta_{4} + \)\(10\!\cdots\!89\)\( \beta_{5} + \)\(12\!\cdots\!64\)\( \beta_{6} + \)\(12\!\cdots\!11\)\( \beta_{7} + \)\(40\!\cdots\!06\)\( \beta_{8} - \)\(35\!\cdots\!27\)\( \beta_{9} + \)\(12\!\cdots\!58\)\( \beta_{10} + \)\(24\!\cdots\!52\)\( \beta_{11} - \)\(29\!\cdots\!30\)\( \beta_{12} - \)\(25\!\cdots\!85\)\( \beta_{13} - \)\(28\!\cdots\!29\)\( \beta_{14} + \)\(11\!\cdots\!03\)\( \beta_{15} + \)\(94\!\cdots\!48\)\( \beta_{16} + \)\(15\!\cdots\!40\)\( \beta_{17} - \)\(31\!\cdots\!76\)\( \beta_{18} - \)\(79\!\cdots\!44\)\( \beta_{19}) q^{97} +(-\)\(22\!\cdots\!56\)\( + \)\(27\!\cdots\!89\)\( \beta_{1} + \)\(56\!\cdots\!10\)\( \beta_{2} + \)\(28\!\cdots\!56\)\( \beta_{3} - \)\(62\!\cdots\!16\)\( \beta_{4} - \)\(17\!\cdots\!12\)\( \beta_{5} + \)\(13\!\cdots\!28\)\( \beta_{6} + \)\(58\!\cdots\!84\)\( \beta_{7} - \)\(21\!\cdots\!10\)\( \beta_{8} + \)\(98\!\cdots\!48\)\( \beta_{9} - \)\(36\!\cdots\!04\)\( \beta_{10} + \)\(11\!\cdots\!38\)\( \beta_{11} - \)\(12\!\cdots\!40\)\( \beta_{12} + \)\(10\!\cdots\!28\)\( \beta_{13} + \)\(40\!\cdots\!86\)\( \beta_{14} + \)\(52\!\cdots\!92\)\( \beta_{15} + \)\(30\!\cdots\!22\)\( \beta_{16} + \)\(18\!\cdots\!12\)\( \beta_{17} + \)\(25\!\cdots\!40\)\( \beta_{18} - \)\(12\!\cdots\!20\)\( \beta_{19}) q^{98} +(\)\(15\!\cdots\!34\)\( - \)\(26\!\cdots\!48\)\( \beta_{1} - \)\(11\!\cdots\!42\)\( \beta_{2} - \)\(22\!\cdots\!97\)\( \beta_{3} + \)\(42\!\cdots\!09\)\( \beta_{4} - \)\(13\!\cdots\!40\)\( \beta_{5} - \)\(10\!\cdots\!12\)\( \beta_{6} + \)\(43\!\cdots\!04\)\( \beta_{7} + \)\(26\!\cdots\!47\)\( \beta_{8} - \)\(14\!\cdots\!32\)\( \beta_{9} - \)\(10\!\cdots\!90\)\( \beta_{10} + \)\(12\!\cdots\!78\)\( \beta_{11} - \)\(18\!\cdots\!59\)\( \beta_{12} + \)\(79\!\cdots\!47\)\( \beta_{13} + \)\(15\!\cdots\!02\)\( \beta_{14} + \)\(55\!\cdots\!44\)\( \beta_{15} + \)\(10\!\cdots\!41\)\( \beta_{16} + \)\(13\!\cdots\!37\)\( \beta_{17} - \)\(88\!\cdots\!30\)\( \beta_{18} - \)\(24\!\cdots\!45\)\( \beta_{19}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 1420757969558292q^{3} - \)\(15\!\cdots\!20\)\(q^{4} - \)\(70\!\cdots\!80\)\(q^{6} + \)\(68\!\cdots\!76\)\(q^{7} - \)\(22\!\cdots\!00\)\(q^{9} + O(q^{10}) \) \( 20q - 1420757969558292q^{3} - \)\(15\!\cdots\!20\)\(q^{4} - \)\(70\!\cdots\!80\)\(q^{6} + \)\(68\!\cdots\!76\)\(q^{7} - \)\(22\!\cdots\!00\)\(q^{9} + \)\(33\!\cdots\!00\)\(q^{10} + \)\(65\!\cdots\!48\)\(q^{12} - \)\(75\!\cdots\!24\)\(q^{13} + \)\(38\!\cdots\!00\)\(q^{15} + \)\(33\!\cdots\!60\)\(q^{16} - \)\(36\!\cdots\!40\)\(q^{18} + \)\(10\!\cdots\!00\)\(q^{19} - \)\(13\!\cdots\!80\)\(q^{21} - \)\(80\!\cdots\!60\)\(q^{22} + \)\(37\!\cdots\!60\)\(q^{24} - \)\(14\!\cdots\!00\)\(q^{25} + \)\(51\!\cdots\!48\)\(q^{27} - \)\(38\!\cdots\!64\)\(q^{28} - \)\(14\!\cdots\!00\)\(q^{30} + \)\(45\!\cdots\!40\)\(q^{31} - \)\(28\!\cdots\!80\)\(q^{33} - \)\(78\!\cdots\!40\)\(q^{34} + \)\(75\!\cdots\!40\)\(q^{36} - \)\(16\!\cdots\!04\)\(q^{37} + \)\(45\!\cdots\!20\)\(q^{39} - \)\(31\!\cdots\!00\)\(q^{40} - \)\(22\!\cdots\!80\)\(q^{42} + \)\(10\!\cdots\!16\)\(q^{43} + \)\(37\!\cdots\!00\)\(q^{45} - \)\(29\!\cdots\!60\)\(q^{46} - \)\(12\!\cdots\!32\)\(q^{48} + \)\(20\!\cdots\!20\)\(q^{49} - \)\(89\!\cdots\!80\)\(q^{51} + \)\(12\!\cdots\!76\)\(q^{52} + \)\(13\!\cdots\!80\)\(q^{54} + \)\(86\!\cdots\!00\)\(q^{55} - \)\(41\!\cdots\!84\)\(q^{57} + \)\(11\!\cdots\!60\)\(q^{58} - \)\(45\!\cdots\!00\)\(q^{60} + \)\(45\!\cdots\!40\)\(q^{61} - \)\(18\!\cdots\!04\)\(q^{63} + \)\(30\!\cdots\!40\)\(q^{64} - \)\(73\!\cdots\!00\)\(q^{66} + \)\(10\!\cdots\!16\)\(q^{67} - \)\(25\!\cdots\!20\)\(q^{69} + \)\(55\!\cdots\!00\)\(q^{70} - \)\(56\!\cdots\!40\)\(q^{72} - \)\(56\!\cdots\!04\)\(q^{73} + \)\(22\!\cdots\!00\)\(q^{75} - \)\(55\!\cdots\!20\)\(q^{76} + \)\(96\!\cdots\!00\)\(q^{78} - \)\(22\!\cdots\!00\)\(q^{79} + \)\(37\!\cdots\!20\)\(q^{81} - \)\(69\!\cdots\!40\)\(q^{82} + \)\(62\!\cdots\!20\)\(q^{84} - \)\(14\!\cdots\!00\)\(q^{85} + \)\(23\!\cdots\!00\)\(q^{87} + \)\(34\!\cdots\!60\)\(q^{88} - \)\(19\!\cdots\!00\)\(q^{90} + \)\(12\!\cdots\!40\)\(q^{91} - \)\(27\!\cdots\!84\)\(q^{93} + \)\(16\!\cdots\!60\)\(q^{94} - \)\(37\!\cdots\!40\)\(q^{96} + \)\(81\!\cdots\!56\)\(q^{97} + \)\(31\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 7311620310544130870 x^{18} + \)\(22\!\cdots\!60\)\( x^{16} + \)\(37\!\cdots\!40\)\( x^{14} + \)\(36\!\cdots\!80\)\( x^{12} + \)\(21\!\cdots\!24\)\( x^{10} + \)\(79\!\cdots\!00\)\( x^{8} + \)\(17\!\cdots\!00\)\( x^{6} + \)\(19\!\cdots\!00\)\( x^{4} + \)\(10\!\cdots\!00\)\( x^{2} + \)\(16\!\cdots\!00\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(13\!\cdots\!75\)\( \nu^{19} - \)\(12\!\cdots\!92\)\( \nu^{18} + \)\(92\!\cdots\!50\)\( \nu^{17} - \)\(89\!\cdots\!40\)\( \nu^{16} + \)\(26\!\cdots\!00\)\( \nu^{15} - \)\(25\!\cdots\!60\)\( \nu^{14} + \)\(41\!\cdots\!00\)\( \nu^{13} - \)\(40\!\cdots\!20\)\( \nu^{12} + \)\(37\!\cdots\!00\)\( \nu^{11} - \)\(36\!\cdots\!00\)\( \nu^{10} + \)\(19\!\cdots\!00\)\( \nu^{9} - \)\(18\!\cdots\!48\)\( \nu^{8} + \)\(55\!\cdots\!00\)\( \nu^{7} - \)\(54\!\cdots\!40\)\( \nu^{6} + \)\(82\!\cdots\!00\)\( \nu^{5} - \)\(80\!\cdots\!00\)\( \nu^{4} + \)\(51\!\cdots\!00\)\( \nu^{3} - \)\(50\!\cdots\!00\)\( \nu^{2} + \)\(85\!\cdots\!00\)\( \nu - \)\(84\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(13\!\cdots\!75\)\( \nu^{19} - \)\(12\!\cdots\!92\)\( \nu^{18} + \)\(92\!\cdots\!50\)\( \nu^{17} - \)\(89\!\cdots\!40\)\( \nu^{16} + \)\(26\!\cdots\!00\)\( \nu^{15} - \)\(25\!\cdots\!60\)\( \nu^{14} + \)\(41\!\cdots\!00\)\( \nu^{13} - \)\(40\!\cdots\!20\)\( \nu^{12} + \)\(37\!\cdots\!00\)\( \nu^{11} - \)\(36\!\cdots\!00\)\( \nu^{10} + \)\(19\!\cdots\!00\)\( \nu^{9} - \)\(18\!\cdots\!48\)\( \nu^{8} + \)\(55\!\cdots\!00\)\( \nu^{7} - \)\(54\!\cdots\!40\)\( \nu^{6} + \)\(82\!\cdots\!00\)\( \nu^{5} - \)\(80\!\cdots\!00\)\( \nu^{4} + \)\(51\!\cdots\!00\)\( \nu^{3} - \)\(49\!\cdots\!00\)\( \nu^{2} + \)\(85\!\cdots\!00\)\( \nu - \)\(39\!\cdots\!00\)\(\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(11\!\cdots\!25\)\( \nu^{19} - \)\(52\!\cdots\!28\)\( \nu^{18} + \)\(81\!\cdots\!50\)\( \nu^{17} - \)\(37\!\cdots\!60\)\( \nu^{16} + \)\(23\!\cdots\!00\)\( \nu^{15} - \)\(10\!\cdots\!40\)\( \nu^{14} + \)\(37\!\cdots\!00\)\( \nu^{13} - \)\(16\!\cdots\!80\)\( \nu^{12} + \)\(33\!\cdots\!00\)\( \nu^{11} - \)\(14\!\cdots\!00\)\( \nu^{10} + \)\(17\!\cdots\!00\)\( \nu^{9} - \)\(77\!\cdots\!32\)\( \nu^{8} + \)\(49\!\cdots\!00\)\( \nu^{7} - \)\(22\!\cdots\!60\)\( \nu^{6} + \)\(73\!\cdots\!00\)\( \nu^{5} - \)\(33\!\cdots\!00\)\( \nu^{4} + \)\(46\!\cdots\!00\)\( \nu^{3} - \)\(20\!\cdots\!00\)\( \nu^{2} + \)\(78\!\cdots\!00\)\( \nu - \)\(34\!\cdots\!00\)\(\)\()/ \)\(95\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(11\!\cdots\!61\)\( \nu^{19} + \)\(29\!\cdots\!40\)\( \nu^{18} + \)\(77\!\cdots\!70\)\( \nu^{17} + \)\(20\!\cdots\!00\)\( \nu^{16} + \)\(22\!\cdots\!80\)\( \nu^{15} + \)\(59\!\cdots\!00\)\( \nu^{14} + \)\(35\!\cdots\!60\)\( \nu^{13} + \)\(91\!\cdots\!00\)\( \nu^{12} + \)\(31\!\cdots\!00\)\( \nu^{11} + \)\(81\!\cdots\!00\)\( \nu^{10} + \)\(16\!\cdots\!84\)\( \nu^{9} + \)\(41\!\cdots\!60\)\( \nu^{8} + \)\(47\!\cdots\!20\)\( \nu^{7} + \)\(12\!\cdots\!00\)\( \nu^{6} + \)\(70\!\cdots\!00\)\( \nu^{5} + \)\(17\!\cdots\!00\)\( \nu^{4} + \)\(44\!\cdots\!00\)\( \nu^{3} + \)\(11\!\cdots\!00\)\( \nu^{2} + \)\(73\!\cdots\!00\)\( \nu + \)\(18\!\cdots\!00\)\(\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(66\!\cdots\!83\)\( \nu^{19} - \)\(90\!\cdots\!68\)\( \nu^{18} - \)\(46\!\cdots\!10\)\( \nu^{17} - \)\(63\!\cdots\!40\)\( \nu^{16} - \)\(13\!\cdots\!40\)\( \nu^{15} - \)\(18\!\cdots\!20\)\( \nu^{14} - \)\(21\!\cdots\!80\)\( \nu^{13} - \)\(28\!\cdots\!60\)\( \nu^{12} - \)\(18\!\cdots\!00\)\( \nu^{11} - \)\(25\!\cdots\!80\)\( \nu^{10} - \)\(98\!\cdots\!52\)\( \nu^{9} - \)\(13\!\cdots\!72\)\( \nu^{8} - \)\(28\!\cdots\!60\)\( \nu^{7} - \)\(38\!\cdots\!60\)\( \nu^{6} - \)\(42\!\cdots\!00\)\( \nu^{5} - \)\(57\!\cdots\!00\)\( \nu^{4} - \)\(26\!\cdots\!00\)\( \nu^{3} - \)\(36\!\cdots\!00\)\( \nu^{2} - \)\(44\!\cdots\!00\)\( \nu - \)\(60\!\cdots\!00\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(18\!\cdots\!09\)\( \nu^{19} + \)\(18\!\cdots\!40\)\( \nu^{18} + \)\(13\!\cdots\!30\)\( \nu^{17} + \)\(13\!\cdots\!00\)\( \nu^{16} + \)\(38\!\cdots\!20\)\( \nu^{15} + \)\(38\!\cdots\!00\)\( \nu^{14} + \)\(59\!\cdots\!40\)\( \nu^{13} + \)\(59\!\cdots\!00\)\( \nu^{12} + \)\(53\!\cdots\!00\)\( \nu^{11} + \)\(52\!\cdots\!00\)\( \nu^{10} + \)\(27\!\cdots\!96\)\( \nu^{9} + \)\(27\!\cdots\!60\)\( \nu^{8} + \)\(79\!\cdots\!80\)\( \nu^{7} + \)\(79\!\cdots\!00\)\( \nu^{6} + \)\(11\!\cdots\!00\)\( \nu^{5} + \)\(11\!\cdots\!00\)\( \nu^{4} + \)\(74\!\cdots\!00\)\( \nu^{3} + \)\(73\!\cdots\!00\)\( \nu^{2} + \)\(12\!\cdots\!00\)\( \nu + \)\(12\!\cdots\!00\)\(\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(10\!\cdots\!89\)\( \nu^{19} - \)\(41\!\cdots\!92\)\( \nu^{18} + \)\(77\!\cdots\!30\)\( \nu^{17} - \)\(29\!\cdots\!40\)\( \nu^{16} + \)\(22\!\cdots\!20\)\( \nu^{15} - \)\(85\!\cdots\!60\)\( \nu^{14} + \)\(34\!\cdots\!40\)\( \nu^{13} - \)\(13\!\cdots\!20\)\( \nu^{12} + \)\(31\!\cdots\!00\)\( \nu^{11} - \)\(11\!\cdots\!00\)\( \nu^{10} + \)\(16\!\cdots\!16\)\( \nu^{9} - \)\(61\!\cdots\!48\)\( \nu^{8} + \)\(46\!\cdots\!80\)\( \nu^{7} - \)\(17\!\cdots\!40\)\( \nu^{6} + \)\(69\!\cdots\!00\)\( \nu^{5} - \)\(26\!\cdots\!00\)\( \nu^{4} + \)\(43\!\cdots\!00\)\( \nu^{3} - \)\(16\!\cdots\!00\)\( \nu^{2} + \)\(73\!\cdots\!00\)\( \nu - \)\(27\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(13\!\cdots\!03\)\( \nu^{19} - \)\(15\!\cdots\!00\)\( \nu^{18} - \)\(96\!\cdots\!90\)\( \nu^{17} - \)\(11\!\cdots\!00\)\( \nu^{16} - \)\(28\!\cdots\!20\)\( \nu^{15} - \)\(33\!\cdots\!00\)\( \nu^{14} - \)\(43\!\cdots\!60\)\( \nu^{13} - \)\(51\!\cdots\!00\)\( \nu^{12} - \)\(39\!\cdots\!80\)\( \nu^{11} - \)\(46\!\cdots\!00\)\( \nu^{10} - \)\(20\!\cdots\!12\)\( \nu^{9} - \)\(23\!\cdots\!00\)\( \nu^{8} - \)\(59\!\cdots\!60\)\( \nu^{7} - \)\(69\!\cdots\!00\)\( \nu^{6} - \)\(88\!\cdots\!00\)\( \nu^{5} - \)\(10\!\cdots\!00\)\( \nu^{4} - \)\(56\!\cdots\!00\)\( \nu^{3} - \)\(65\!\cdots\!00\)\( \nu^{2} - \)\(97\!\cdots\!00\)\( \nu - \)\(10\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(24\!\cdots\!85\)\( \nu^{19} + \)\(22\!\cdots\!88\)\( \nu^{18} + \)\(17\!\cdots\!30\)\( \nu^{17} + \)\(15\!\cdots\!00\)\( \nu^{16} + \)\(49\!\cdots\!80\)\( \nu^{15} + \)\(45\!\cdots\!80\)\( \nu^{14} + \)\(77\!\cdots\!80\)\( \nu^{13} + \)\(71\!\cdots\!20\)\( \nu^{12} + \)\(69\!\cdots\!80\)\( \nu^{11} + \)\(63\!\cdots\!40\)\( \nu^{10} + \)\(36\!\cdots\!20\)\( \nu^{9} + \)\(32\!\cdots\!12\)\( \nu^{8} + \)\(10\!\cdots\!00\)\( \nu^{7} + \)\(95\!\cdots\!60\)\( \nu^{6} + \)\(15\!\cdots\!00\)\( \nu^{5} + \)\(14\!\cdots\!00\)\( \nu^{4} + \)\(98\!\cdots\!00\)\( \nu^{3} + \)\(88\!\cdots\!00\)\( \nu^{2} + \)\(16\!\cdots\!00\)\( \nu + \)\(14\!\cdots\!00\)\(\)\()/ \)\(95\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(37\!\cdots\!67\)\( \nu^{19} + \)\(12\!\cdots\!68\)\( \nu^{18} + \)\(26\!\cdots\!90\)\( \nu^{17} + \)\(89\!\cdots\!00\)\( \nu^{16} + \)\(76\!\cdots\!60\)\( \nu^{15} + \)\(25\!\cdots\!80\)\( \nu^{14} + \)\(11\!\cdots\!20\)\( \nu^{13} + \)\(39\!\cdots\!20\)\( \nu^{12} + \)\(10\!\cdots\!00\)\( \nu^{11} + \)\(35\!\cdots\!40\)\( \nu^{10} + \)\(55\!\cdots\!48\)\( \nu^{9} + \)\(18\!\cdots\!32\)\( \nu^{8} + \)\(15\!\cdots\!40\)\( \nu^{7} + \)\(52\!\cdots\!60\)\( \nu^{6} + \)\(23\!\cdots\!00\)\( \nu^{5} + \)\(77\!\cdots\!00\)\( \nu^{4} + \)\(14\!\cdots\!00\)\( \nu^{3} + \)\(48\!\cdots\!00\)\( \nu^{2} + \)\(24\!\cdots\!00\)\( \nu + \)\(81\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(31\!\cdots\!49\)\( \nu^{19} - \)\(89\!\cdots\!92\)\( \nu^{18} + \)\(22\!\cdots\!10\)\( \nu^{17} - \)\(62\!\cdots\!20\)\( \nu^{16} + \)\(64\!\cdots\!00\)\( \nu^{15} - \)\(18\!\cdots\!40\)\( \nu^{14} + \)\(10\!\cdots\!20\)\( \nu^{13} - \)\(28\!\cdots\!00\)\( \nu^{12} + \)\(89\!\cdots\!80\)\( \nu^{11} - \)\(25\!\cdots\!80\)\( \nu^{10} + \)\(46\!\cdots\!36\)\( \nu^{9} - \)\(12\!\cdots\!28\)\( \nu^{8} + \)\(13\!\cdots\!80\)\( \nu^{7} - \)\(37\!\cdots\!40\)\( \nu^{6} + \)\(19\!\cdots\!00\)\( \nu^{5} - \)\(55\!\cdots\!00\)\( \nu^{4} + \)\(12\!\cdots\!00\)\( \nu^{3} - \)\(34\!\cdots\!00\)\( \nu^{2} + \)\(20\!\cdots\!00\)\( \nu - \)\(58\!\cdots\!00\)\(\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(57\!\cdots\!81\)\( \nu^{19} + \)\(79\!\cdots\!20\)\( \nu^{18} - \)\(41\!\cdots\!50\)\( \nu^{17} + \)\(56\!\cdots\!20\)\( \nu^{16} - \)\(11\!\cdots\!60\)\( \nu^{15} + \)\(16\!\cdots\!20\)\( \nu^{14} - \)\(18\!\cdots\!40\)\( \nu^{13} + \)\(25\!\cdots\!20\)\( \nu^{12} - \)\(16\!\cdots\!80\)\( \nu^{11} + \)\(22\!\cdots\!20\)\( \nu^{10} - \)\(86\!\cdots\!44\)\( \nu^{9} + \)\(11\!\cdots\!00\)\( \nu^{8} - \)\(25\!\cdots\!20\)\( \nu^{7} + \)\(34\!\cdots\!00\)\( \nu^{6} - \)\(37\!\cdots\!00\)\( \nu^{5} + \)\(51\!\cdots\!00\)\( \nu^{4} - \)\(24\!\cdots\!00\)\( \nu^{3} + \)\(32\!\cdots\!00\)\( \nu^{2} - \)\(41\!\cdots\!00\)\( \nu + \)\(53\!\cdots\!00\)\(\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(21\!\cdots\!29\)\( \nu^{19} + \)\(19\!\cdots\!40\)\( \nu^{18} - \)\(14\!\cdots\!70\)\( \nu^{17} + \)\(13\!\cdots\!60\)\( \nu^{16} - \)\(43\!\cdots\!60\)\( \nu^{15} + \)\(39\!\cdots\!60\)\( \nu^{14} - \)\(68\!\cdots\!80\)\( \nu^{13} + \)\(61\!\cdots\!60\)\( \nu^{12} - \)\(61\!\cdots\!40\)\( \nu^{11} + \)\(54\!\cdots\!60\)\( \nu^{10} - \)\(31\!\cdots\!16\)\( \nu^{9} + \)\(28\!\cdots\!20\)\( \nu^{8} - \)\(91\!\cdots\!80\)\( \nu^{7} + \)\(82\!\cdots\!00\)\( \nu^{6} - \)\(13\!\cdots\!00\)\( \nu^{5} + \)\(12\!\cdots\!00\)\( \nu^{4} - \)\(85\!\cdots\!00\)\( \nu^{3} + \)\(76\!\cdots\!00\)\( \nu^{2} - \)\(13\!\cdots\!00\)\( \nu + \)\(12\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(60\!\cdots\!59\)\( \nu^{19} + \)\(72\!\cdots\!28\)\( \nu^{18} - \)\(43\!\cdots\!70\)\( \nu^{17} + \)\(51\!\cdots\!80\)\( \nu^{16} - \)\(12\!\cdots\!60\)\( \nu^{15} + \)\(14\!\cdots\!60\)\( \nu^{14} - \)\(19\!\cdots\!80\)\( \nu^{13} + \)\(23\!\cdots\!00\)\( \nu^{12} - \)\(17\!\cdots\!40\)\( \nu^{11} + \)\(20\!\cdots\!20\)\( \nu^{10} - \)\(91\!\cdots\!36\)\( \nu^{9} + \)\(10\!\cdots\!52\)\( \nu^{8} - \)\(26\!\cdots\!80\)\( \nu^{7} + \)\(31\!\cdots\!60\)\( \nu^{6} - \)\(40\!\cdots\!00\)\( \nu^{5} + \)\(46\!\cdots\!00\)\( \nu^{4} - \)\(25\!\cdots\!00\)\( \nu^{3} + \)\(29\!\cdots\!00\)\( \nu^{2} - \)\(44\!\cdots\!00\)\( \nu + \)\(49\!\cdots\!00\)\(\)\()/ \)\(38\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(20\!\cdots\!81\)\( \nu^{19} + \)\(16\!\cdots\!08\)\( \nu^{18} + \)\(14\!\cdots\!90\)\( \nu^{17} + \)\(11\!\cdots\!20\)\( \nu^{16} + \)\(42\!\cdots\!00\)\( \nu^{15} + \)\(33\!\cdots\!00\)\( \nu^{14} + \)\(65\!\cdots\!80\)\( \nu^{13} + \)\(51\!\cdots\!40\)\( \nu^{12} + \)\(58\!\cdots\!20\)\( \nu^{11} + \)\(46\!\cdots\!60\)\( \nu^{10} + \)\(30\!\cdots\!84\)\( \nu^{9} + \)\(24\!\cdots\!12\)\( \nu^{8} + \)\(88\!\cdots\!20\)\( \nu^{7} + \)\(69\!\cdots\!60\)\( \nu^{6} + \)\(13\!\cdots\!00\)\( \nu^{5} + \)\(10\!\cdots\!00\)\( \nu^{4} + \)\(82\!\cdots\!00\)\( \nu^{3} + \)\(65\!\cdots\!00\)\( \nu^{2} + \)\(13\!\cdots\!00\)\( \nu + \)\(10\!\cdots\!00\)\(\)\()/ \)\(33\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(59\!\cdots\!53\)\( \nu^{19} - \)\(60\!\cdots\!04\)\( \nu^{18} - \)\(41\!\cdots\!90\)\( \nu^{17} - \)\(42\!\cdots\!60\)\( \nu^{16} - \)\(12\!\cdots\!20\)\( \nu^{15} - \)\(12\!\cdots\!00\)\( \nu^{14} - \)\(19\!\cdots\!60\)\( \nu^{13} - \)\(19\!\cdots\!20\)\( \nu^{12} - \)\(17\!\cdots\!80\)\( \nu^{11} - \)\(17\!\cdots\!80\)\( \nu^{10} - \)\(88\!\cdots\!12\)\( \nu^{9} - \)\(88\!\cdots\!56\)\( \nu^{8} - \)\(25\!\cdots\!60\)\( \nu^{7} - \)\(25\!\cdots\!80\)\( \nu^{6} - \)\(38\!\cdots\!00\)\( \nu^{5} - \)\(38\!\cdots\!00\)\( \nu^{4} - \)\(23\!\cdots\!00\)\( \nu^{3} - \)\(24\!\cdots\!00\)\( \nu^{2} - \)\(38\!\cdots\!00\)\( \nu - \)\(39\!\cdots\!00\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(22\!\cdots\!13\)\( \nu^{19} + \)\(38\!\cdots\!12\)\( \nu^{18} + \)\(15\!\cdots\!50\)\( \nu^{17} + \)\(26\!\cdots\!60\)\( \nu^{16} + \)\(45\!\cdots\!80\)\( \nu^{15} + \)\(78\!\cdots\!80\)\( \nu^{14} + \)\(70\!\cdots\!20\)\( \nu^{13} + \)\(12\!\cdots\!40\)\( \nu^{12} + \)\(63\!\cdots\!40\)\( \nu^{11} + \)\(10\!\cdots\!20\)\( \nu^{10} + \)\(32\!\cdots\!12\)\( \nu^{9} + \)\(56\!\cdots\!48\)\( \nu^{8} + \)\(95\!\cdots\!60\)\( \nu^{7} + \)\(16\!\cdots\!40\)\( \nu^{6} + \)\(14\!\cdots\!00\)\( \nu^{5} + \)\(24\!\cdots\!00\)\( \nu^{4} + \)\(89\!\cdots\!00\)\( \nu^{3} + \)\(15\!\cdots\!00\)\( \nu^{2} + \)\(15\!\cdots\!00\)\( \nu + \)\(26\!\cdots\!00\)\(\)\()/ \)\(57\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(17\!\cdots\!79\)\( \nu^{19} + \)\(74\!\cdots\!28\)\( \nu^{18} - \)\(12\!\cdots\!30\)\( \nu^{17} + \)\(53\!\cdots\!60\)\( \nu^{16} - \)\(35\!\cdots\!20\)\( \nu^{15} + \)\(15\!\cdots\!40\)\( \nu^{14} - \)\(55\!\cdots\!40\)\( \nu^{13} + \)\(24\!\cdots\!80\)\( \nu^{12} - \)\(49\!\cdots\!00\)\( \nu^{11} + \)\(21\!\cdots\!00\)\( \nu^{10} - \)\(25\!\cdots\!76\)\( \nu^{9} + \)\(11\!\cdots\!32\)\( \nu^{8} - \)\(74\!\cdots\!80\)\( \nu^{7} + \)\(32\!\cdots\!60\)\( \nu^{6} - \)\(11\!\cdots\!00\)\( \nu^{5} + \)\(48\!\cdots\!00\)\( \nu^{4} - \)\(70\!\cdots\!00\)\( \nu^{3} + \)\(31\!\cdots\!00\)\( \nu^{2} - \)\(11\!\cdots\!00\)\( \nu + \)\(53\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 68 \beta_{2} - 34 \beta_{1} - 26321833117958871105\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{8} - 5 \beta_{7} - 22 \beta_{5} + 80800 \beta_{4} - 2512797 \beta_{3} - 4793992141335 \beta_{2} - 43046734616059862085 \beta_{1} + 1917597342933\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{19} + 4 \beta_{18} - 2 \beta_{16} - \beta_{15} - 384 \beta_{14} - 51 \beta_{13} - 3498 \beta_{12} + 3509 \beta_{11} - 173175 \beta_{10} + 298 \beta_{9} + 2121544 \beta_{8} + 1441158400 \beta_{7} + 45368758878 \beta_{6} + 5127411236751 \beta_{5} - 4532452580139 \beta_{4} - 58389972698826504858 \beta_{3} + 2250733732529448566510 \beta_{2} + 1132864548727007224065 \beta_{1} + 1133068902293810336812765305862281710515\)\()/1296\)
\(\nu^{5}\)\(=\)\((\)\(162140790 \beta_{19} - 324281580 \beta_{18} + 492987401 \beta_{17} - 7667037987 \beta_{16} - 4609393797 \beta_{15} + 1185831794550 \beta_{14} - 3130431594 \beta_{13} + 1427117764195 \beta_{12} + 20134128198573 \beta_{11} + 51691800462691 \beta_{10} - 7977138499204606 \beta_{9} - 8148944670240232868 \beta_{8} + 54792885476885982438 \beta_{7} - 897550654872864596 \beta_{6} + 1310766662167743483113 \beta_{5} - 1531788619846931112696939 \beta_{4} + 26917238563473982311318781 \beta_{3} + 51686376372715299204711229310803 \beta_{2} + 263702532328848915556361040911612542147 \beta_{1} - 20674555626172489806311078032392\)\()/972\)
\(\nu^{6}\)\(=\)\((\)\(-9275871270697541955 \beta_{19} - 37103485082790167820 \beta_{18} - 433249668314838576 \beta_{17} + 18984992209709922486 \beta_{16} + 7042881698946977619 \beta_{15} + 4487888652654099128928 \beta_{14} + 229077539648869822425 \beta_{13} + 49462848776426456662382 \beta_{12} - 49478161002232669778351 \beta_{11} + 3965320631765348398678293 \beta_{10} + 19318924274650254774315202 \beta_{9} - 33684729924700945481137864 \beta_{8} - 45493212968027783064815652720 \beta_{7} - 655343895772016568439306348346 \beta_{6} - 76181998863787133702752756418365 \beta_{5} + 122290029493680967779222799458865 \beta_{4} + 389756860802480553672386479787308425390 \beta_{3} + 18399208390928271543874294248359353110550 \beta_{2} + 9003843615988274364335588552785155462493 \beta_{1} - 6941133374434453974974455422625863747185510944844500422953\)\()/5832\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(11\!\cdots\!10\)\( \beta_{19} + \)\(23\!\cdots\!20\)\( \beta_{18} - \)\(62\!\cdots\!59\)\( \beta_{17} + \)\(10\!\cdots\!09\)\( \beta_{16} + \)\(47\!\cdots\!39\)\( \beta_{15} - \)\(14\!\cdots\!54\)\( \beta_{14} + \)\(28\!\cdots\!62\)\( \beta_{13} - \)\(18\!\cdots\!85\)\( \beta_{12} - \)\(21\!\cdots\!75\)\( \beta_{11} - \)\(63\!\cdots\!25\)\( \beta_{10} + \)\(87\!\cdots\!74\)\( \beta_{9} + \)\(56\!\cdots\!16\)\( \beta_{8} - \)\(51\!\cdots\!02\)\( \beta_{7} + \)\(13\!\cdots\!44\)\( \beta_{6} - \)\(17\!\cdots\!95\)\( \beta_{5} + \)\(22\!\cdots\!73\)\( \beta_{4} - \)\(24\!\cdots\!27\)\( \beta_{3} - \)\(48\!\cdots\!65\)\( \beta_{2} - \)\(16\!\cdots\!93\)\( \beta_{1} + \)\(19\!\cdots\!56\)\(\)\()/4374\)
\(\nu^{8}\)\(=\)\((\)\(\)\(23\!\cdots\!15\)\( \beta_{19} + \)\(92\!\cdots\!60\)\( \beta_{18} + \)\(25\!\cdots\!56\)\( \beta_{17} - \)\(48\!\cdots\!86\)\( \beta_{16} + \)\(11\!\cdots\!85\)\( \beta_{15} - \)\(14\!\cdots\!88\)\( \beta_{14} - \)\(32\!\cdots\!01\)\( \beta_{13} - \)\(16\!\cdots\!98\)\( \beta_{12} + \)\(16\!\cdots\!03\)\( \beta_{11} - \)\(16\!\cdots\!33\)\( \beta_{10} - \)\(13\!\cdots\!50\)\( \beta_{9} + \)\(12\!\cdots\!72\)\( \beta_{8} + \)\(19\!\cdots\!60\)\( \beta_{7} + \)\(21\!\cdots\!46\)\( \beta_{6} + \)\(26\!\cdots\!61\)\( \beta_{5} - \)\(50\!\cdots\!61\)\( \beta_{4} - \)\(85\!\cdots\!50\)\( \beta_{3} - \)\(12\!\cdots\!30\)\( \beta_{2} - \)\(62\!\cdots\!13\)\( \beta_{1} + \)\(14\!\cdots\!17\)\(\)\()/8748\)
\(\nu^{9}\)\(=\)\((\)\(\)\(16\!\cdots\!30\)\( \beta_{19} - \)\(33\!\cdots\!60\)\( \beta_{18} + \)\(20\!\cdots\!15\)\( \beta_{17} - \)\(37\!\cdots\!61\)\( \beta_{16} - \)\(12\!\cdots\!51\)\( \beta_{15} + \)\(48\!\cdots\!74\)\( \beta_{14} - \)\(62\!\cdots\!06\)\( \beta_{13} + \)\(61\!\cdots\!45\)\( \beta_{12} + \)\(59\!\cdots\!43\)\( \beta_{11} + \)\(19\!\cdots\!21\)\( \beta_{10} - \)\(24\!\cdots\!70\)\( \beta_{9} - \)\(12\!\cdots\!64\)\( \beta_{8} + \)\(14\!\cdots\!78\)\( \beta_{7} - \)\(44\!\cdots\!20\)\( \beta_{6} + \)\(59\!\cdots\!47\)\( \beta_{5} - \)\(81\!\cdots\!57\)\( \beta_{4} + \)\(70\!\cdots\!39\)\( \beta_{3} + \)\(13\!\cdots\!45\)\( \beta_{2} + \)\(36\!\cdots\!57\)\( \beta_{1} - \)\(54\!\cdots\!84\)\(\)\()/6561\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(16\!\cdots\!05\)\( \beta_{19} - \)\(65\!\cdots\!20\)\( \beta_{18} - \)\(28\!\cdots\!84\)\( \beta_{17} + \)\(35\!\cdots\!94\)\( \beta_{16} - \)\(33\!\cdots\!31\)\( \beta_{15} + \)\(12\!\cdots\!32\)\( \beta_{14} + \)\(17\!\cdots\!23\)\( \beta_{13} + \)\(14\!\cdots\!78\)\( \beta_{12} - \)\(14\!\cdots\!17\)\( \beta_{11} + \)\(16\!\cdots\!43\)\( \beta_{10} + \)\(16\!\cdots\!42\)\( \beta_{9} - \)\(11\!\cdots\!44\)\( \beta_{8} - \)\(19\!\cdots\!72\)\( \beta_{7} - \)\(19\!\cdots\!06\)\( \beta_{6} - \)\(23\!\cdots\!87\)\( \beta_{5} + \)\(49\!\cdots\!55\)\( \beta_{4} + \)\(56\!\cdots\!62\)\( \beta_{3} + \)\(14\!\cdots\!86\)\( \beta_{2} + \)\(70\!\cdots\!95\)\( \beta_{1} - \)\(95\!\cdots\!27\)\(\)\()/39366\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(18\!\cdots\!00\)\( \beta_{19} + \)\(36\!\cdots\!00\)\( \beta_{18} - \)\(35\!\cdots\!86\)\( \beta_{17} + \)\(68\!\cdots\!02\)\( \beta_{16} + \)\(17\!\cdots\!62\)\( \beta_{15} - \)\(83\!\cdots\!80\)\( \beta_{14} + \)\(70\!\cdots\!04\)\( \beta_{13} - \)\(10\!\cdots\!50\)\( \beta_{12} - \)\(93\!\cdots\!18\)\( \beta_{11} - \)\(32\!\cdots\!86\)\( \beta_{10} + \)\(37\!\cdots\!36\)\( \beta_{9} + \)\(17\!\cdots\!08\)\( \beta_{8} - \)\(23\!\cdots\!12\)\( \beta_{7} + \)\(80\!\cdots\!56\)\( \beta_{6} - \)\(10\!\cdots\!30\)\( \beta_{5} + \)\(15\!\cdots\!74\)\( \beta_{4} - \)\(11\!\cdots\!54\)\( \beta_{3} - \)\(22\!\cdots\!14\)\( \beta_{2} - \)\(47\!\cdots\!18\)\( \beta_{1} + \)\(88\!\cdots\!00\)\(\)\()/59049\)
\(\nu^{12}\)\(=\)\((\)\(\)\(38\!\cdots\!05\)\( \beta_{19} + \)\(15\!\cdots\!20\)\( \beta_{18} + \)\(87\!\cdots\!16\)\( \beta_{17} - \)\(85\!\cdots\!26\)\( \beta_{16} + \)\(14\!\cdots\!11\)\( \beta_{15} - \)\(32\!\cdots\!48\)\( \beta_{14} - \)\(37\!\cdots\!35\)\( \beta_{13} - \)\(40\!\cdots\!18\)\( \beta_{12} + \)\(39\!\cdots\!57\)\( \beta_{11} - \)\(49\!\cdots\!91\)\( \beta_{10} - \)\(56\!\cdots\!02\)\( \beta_{9} + \)\(33\!\cdots\!36\)\( \beta_{8} + \)\(57\!\cdots\!56\)\( \beta_{7} + \)\(53\!\cdots\!82\)\( \beta_{6} + \)\(66\!\cdots\!47\)\( \beta_{5} - \)\(14\!\cdots\!11\)\( \beta_{4} - \)\(12\!\cdots\!06\)\( \beta_{3} - \)\(44\!\cdots\!98\)\( \beta_{2} - \)\(21\!\cdots\!47\)\( \beta_{1} + \)\(20\!\cdots\!47\)\(\)\()/59049\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(93\!\cdots\!60\)\( \beta_{19} + \)\(18\!\cdots\!20\)\( \beta_{18} + \)\(19\!\cdots\!84\)\( \beta_{17} - \)\(38\!\cdots\!24\)\( \beta_{16} - \)\(82\!\cdots\!04\)\( \beta_{15} + \)\(46\!\cdots\!64\)\( \beta_{14} - \)\(23\!\cdots\!52\)\( \beta_{13} + \)\(57\!\cdots\!00\)\( \beta_{12} + \)\(48\!\cdots\!60\)\( \beta_{11} + \)\(16\!\cdots\!60\)\( \beta_{10} - \)\(18\!\cdots\!84\)\( \beta_{9} - \)\(77\!\cdots\!36\)\( \beta_{8} + \)\(12\!\cdots\!88\)\( \beta_{7} - \)\(44\!\cdots\!44\)\( \beta_{6} + \)\(57\!\cdots\!28\)\( \beta_{5} - \)\(89\!\cdots\!68\)\( \beta_{4} + \)\(59\!\cdots\!44\)\( \beta_{3} + \)\(11\!\cdots\!44\)\( \beta_{2} + \)\(20\!\cdots\!72\)\( \beta_{1} - \)\(46\!\cdots\!48\)\(\)\()/177147\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(59\!\cdots\!30\)\( \beta_{19} - \)\(23\!\cdots\!20\)\( \beta_{18} - \)\(16\!\cdots\!76\)\( \beta_{17} + \)\(13\!\cdots\!36\)\( \beta_{16} - \)\(30\!\cdots\!30\)\( \beta_{15} + \)\(56\!\cdots\!08\)\( \beta_{14} + \)\(60\!\cdots\!26\)\( \beta_{13} + \)\(72\!\cdots\!76\)\( \beta_{12} - \)\(69\!\cdots\!86\)\( \beta_{11} + \)\(93\!\cdots\!62\)\( \beta_{10} + \)\(11\!\cdots\!20\)\( \beta_{9} - \)\(60\!\cdots\!28\)\( \beta_{8} - \)\(10\!\cdots\!88\)\( \beta_{7} - \)\(94\!\cdots\!44\)\( \beta_{6} - \)\(11\!\cdots\!42\)\( \beta_{5} + \)\(26\!\cdots\!34\)\( \beta_{4} + \)\(18\!\cdots\!88\)\( \beta_{3} + \)\(83\!\cdots\!84\)\( \beta_{2} + \)\(41\!\cdots\!30\)\( \beta_{1} - \)\(30\!\cdots\!74\)\(\)\()/59049\)
\(\nu^{15}\)\(=\)\((\)\(\)\(29\!\cdots\!80\)\( \beta_{19} - \)\(59\!\cdots\!60\)\( \beta_{18} - \)\(34\!\cdots\!44\)\( \beta_{17} + \)\(68\!\cdots\!28\)\( \beta_{16} + \)\(12\!\cdots\!68\)\( \beta_{15} - \)\(81\!\cdots\!00\)\( \beta_{14} + \)\(23\!\cdots\!36\)\( \beta_{13} - \)\(10\!\cdots\!60\)\( \beta_{12} - \)\(81\!\cdots\!92\)\( \beta_{11} - \)\(28\!\cdots\!44\)\( \beta_{10} + \)\(30\!\cdots\!84\)\( \beta_{9} + \)\(11\!\cdots\!52\)\( \beta_{8} - \)\(21\!\cdots\!52\)\( \beta_{7} + \)\(77\!\cdots\!24\)\( \beta_{6} - \)\(10\!\cdots\!12\)\( \beta_{5} + \)\(16\!\cdots\!96\)\( \beta_{4} - \)\(10\!\cdots\!84\)\( \beta_{3} - \)\(19\!\cdots\!92\)\( \beta_{2} - \)\(30\!\cdots\!28\)\( \beta_{1} + \)\(79\!\cdots\!68\)\(\)\()/177147\)
\(\nu^{16}\)\(=\)\((\)\(\)\(93\!\cdots\!00\)\( \beta_{19} + \)\(37\!\cdots\!00\)\( \beta_{18} + \)\(28\!\cdots\!68\)\( \beta_{17} - \)\(21\!\cdots\!68\)\( \beta_{16} + \)\(60\!\cdots\!48\)\( \beta_{15} - \)\(93\!\cdots\!44\)\( \beta_{14} - \)\(10\!\cdots\!60\)\( \beta_{13} - \)\(12\!\cdots\!96\)\( \beta_{12} + \)\(12\!\cdots\!48\)\( \beta_{11} - \)\(16\!\cdots\!44\)\( \beta_{10} - \)\(20\!\cdots\!56\)\( \beta_{9} + \)\(10\!\cdots\!72\)\( \beta_{8} + \)\(18\!\cdots\!00\)\( \beta_{7} + \)\(16\!\cdots\!48\)\( \beta_{6} + \)\(20\!\cdots\!60\)\( \beta_{5} - \)\(45\!\cdots\!00\)\( \beta_{4} - \)\(27\!\cdots\!40\)\( \beta_{3} - \)\(14\!\cdots\!20\)\( \beta_{2} - \)\(73\!\cdots\!84\)\( \beta_{1} + \)\(45\!\cdots\!64\)\(\)\()/59049\)
\(\nu^{17}\)\(=\)\((\)\(-\)\(68\!\cdots\!80\)\( \beta_{19} + \)\(13\!\cdots\!60\)\( \beta_{18} + \)\(60\!\cdots\!08\)\( \beta_{17} - \)\(11\!\cdots\!48\)\( \beta_{16} - \)\(19\!\cdots\!08\)\( \beta_{15} + \)\(14\!\cdots\!08\)\( \beta_{14} - \)\(15\!\cdots\!84\)\( \beta_{13} + \)\(16\!\cdots\!20\)\( \beta_{12} + \)\(13\!\cdots\!20\)\( \beta_{11} + \)\(47\!\cdots\!40\)\( \beta_{10} - \)\(48\!\cdots\!88\)\( \beta_{9} - \)\(17\!\cdots\!52\)\( \beta_{8} + \)\(36\!\cdots\!64\)\( \beta_{7} - \)\(13\!\cdots\!28\)\( \beta_{6} + \)\(16\!\cdots\!80\)\( \beta_{5} - \)\(28\!\cdots\!56\)\( \beta_{4} + \)\(16\!\cdots\!24\)\( \beta_{3} + \)\(33\!\cdots\!60\)\( \beta_{2} + \)\(45\!\cdots\!76\)\( \beta_{1} - \)\(13\!\cdots\!72\)\(\)\()/177147\)
\(\nu^{18}\)\(=\)\((\)\(-\)\(48\!\cdots\!60\)\( \beta_{19} - \)\(19\!\cdots\!40\)\( \beta_{18} - \)\(15\!\cdots\!44\)\( \beta_{17} + \)\(11\!\cdots\!64\)\( \beta_{16} - \)\(37\!\cdots\!00\)\( \beta_{15} + \)\(51\!\cdots\!12\)\( \beta_{14} + \)\(56\!\cdots\!64\)\( \beta_{13} + \)\(70\!\cdots\!32\)\( \beta_{12} - \)\(68\!\cdots\!92\)\( \beta_{11} + \)\(97\!\cdots\!12\)\( \beta_{10} + \)\(12\!\cdots\!20\)\( \beta_{9} - \)\(61\!\cdots\!28\)\( \beta_{8} - \)\(10\!\cdots\!80\)\( \beta_{7} - \)\(92\!\cdots\!44\)\( \beta_{6} - \)\(11\!\cdots\!84\)\( \beta_{5} + \)\(25\!\cdots\!44\)\( \beta_{4} + \)\(13\!\cdots\!40\)\( \beta_{3} + \)\(84\!\cdots\!40\)\( \beta_{2} + \)\(41\!\cdots\!72\)\( \beta_{1} - \)\(22\!\cdots\!28\)\(\)\()/19683\)
\(\nu^{19}\)\(=\)\((\)\(\)\(45\!\cdots\!20\)\( \beta_{19} - \)\(91\!\cdots\!40\)\( \beta_{18} - \)\(34\!\cdots\!20\)\( \beta_{17} + \)\(68\!\cdots\!96\)\( \beta_{16} + \)\(10\!\cdots\!36\)\( \beta_{15} - \)\(79\!\cdots\!24\)\( \beta_{14} - \)\(12\!\cdots\!24\)\( \beta_{13} - \)\(93\!\cdots\!20\)\( \beta_{12} - \)\(75\!\cdots\!08\)\( \beta_{11} - \)\(25\!\cdots\!76\)\( \beta_{10} + \)\(26\!\cdots\!00\)\( \beta_{9} + \)\(88\!\cdots\!04\)\( \beta_{8} - \)\(20\!\cdots\!48\)\( \beta_{7} + \)\(73\!\cdots\!00\)\( \beta_{6} - \)\(93\!\cdots\!52\)\( \beta_{5} + \)\(16\!\cdots\!52\)\( \beta_{4} - \)\(94\!\cdots\!04\)\( \beta_{3} - \)\(18\!\cdots\!80\)\( \beta_{2} - \)\(22\!\cdots\!12\)\( \beta_{1} + \)\(74\!\cdots\!04\)\(\)\()/59049\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
1.25990e9i
1.17774e9i
1.16631e9i
1.15913e9i
7.41715e8i
6.71940e8i
5.00631e8i
4.99832e8i
3.25526e8i
1.58953e8i
1.58953e8i
3.25526e8i
4.99832e8i
5.00631e8i
6.71940e8i
7.41715e8i
1.15913e9i
1.16631e9i
1.17774e9i
1.25990e9i
7.55940e9i −2.98618e14 1.82880e15i −3.86978e19 3.78912e22i −1.38246e25 + 2.25737e24i 1.61394e27 1.53086e29i −3.25534e30 + 1.09222e30i 2.86435e32
2.2 7.06644e9i −1.85210e15 5.82746e13i −3.14879e19 1.76003e22i −4.11794e23 + 1.30878e25i −6.88028e26 9.21545e28i 3.42689e30 + 2.15861e29i −1.24372e32
2.3 6.99784e9i 3.59786e14 + 1.81776e15i −3.05230e19 1.06729e22i 1.27204e25 2.51773e24i 9.75273e26 8.45080e28i −3.17479e30 + 1.30801e30i −7.46870e31
2.4 6.95479e9i 1.80867e15 4.02985e14i −2.99223e19 4.26608e21i −2.80268e24 1.25789e25i −1.59304e27 7.98101e28i 3.10889e30 1.45773e30i −2.96697e31
2.5 4.45029e9i −9.93580e14 + 1.56412e15i −1.35835e18 4.47337e22i 6.96081e24 + 4.42172e24i −9.42280e26 7.60483e28i −1.45928e30 3.10816e30i 1.99078e32
2.6 4.03164e9i 5.57989e14 1.76701e15i 2.19261e18 2.91915e22i −7.12396e24 2.24961e24i 6.92352e26 8.32105e28i −2.81098e30 1.97195e30i −1.17690e32
2.7 3.00379e9i 1.81065e15 + 3.93978e14i 9.42400e18 2.20086e22i 1.18343e24 5.43882e24i 9.43770e26 8.37178e28i 3.12325e30 + 1.42671e30i 6.61093e31
2.8 2.99899e9i −1.38295e15 1.23335e15i 9.45278e18 5.43489e21i −3.69880e24 + 4.14745e24i −9.58380e26 8.36705e28i 3.91402e29 + 3.41130e30i 1.62992e31
2.9 1.95316e9i −1.56985e15 + 9.84500e14i 1.46319e19 1.66808e22i 1.92288e24 + 3.06617e24i 1.45784e27 6.46078e28i 1.49520e30 3.09104e30i −3.25803e31
2.10 9.53717e8i 8.49626e14 + 1.64676e15i 1.75372e19 2.45445e22i 1.57054e24 8.10302e23i −1.16039e27 3.43185e28i −1.98996e30 + 2.79826e30i −2.34085e31
2.11 9.53717e8i 8.49626e14 1.64676e15i 1.75372e19 2.45445e22i 1.57054e24 + 8.10302e23i −1.16039e27 3.43185e28i −1.98996e30 2.79826e30i −2.34085e31
2.12 1.95316e9i −1.56985e15 9.84500e14i 1.46319e19 1.66808e22i 1.92288e24 3.06617e24i 1.45784e27 6.46078e28i 1.49520e30 + 3.09104e30i −3.25803e31
2.13 2.99899e9i −1.38295e15 + 1.23335e15i 9.45278e18 5.43489e21i −3.69880e24 4.14745e24i −9.58380e26 8.36705e28i 3.91402e29 3.41130e30i 1.62992e31
2.14 3.00379e9i 1.81065e15 3.93978e14i 9.42400e18 2.20086e22i 1.18343e24 + 5.43882e24i 9.43770e26 8.37178e28i 3.12325e30 1.42671e30i 6.61093e31
2.15 4.03164e9i 5.57989e14 + 1.76701e15i 2.19261e18 2.91915e22i −7.12396e24 + 2.24961e24i 6.92352e26 8.32105e28i −2.81098e30 + 1.97195e30i −1.17690e32
2.16 4.45029e9i −9.93580e14 1.56412e15i −1.35835e18 4.47337e22i 6.96081e24 4.42172e24i −9.42280e26 7.60483e28i −1.45928e30 + 3.10816e30i 1.99078e32
2.17 6.95479e9i 1.80867e15 + 4.02985e14i −2.99223e19 4.26608e21i −2.80268e24 + 1.25789e25i −1.59304e27 7.98101e28i 3.10889e30 + 1.45773e30i −2.96697e31
2.18 6.99784e9i 3.59786e14 1.81776e15i −3.05230e19 1.06729e22i 1.27204e25 + 2.51773e24i 9.75273e26 8.45080e28i −3.17479e30 1.30801e30i −7.46870e31
2.19 7.06644e9i −1.85210e15 + 5.82746e13i −3.14879e19 1.76003e22i −4.11794e23 1.30878e25i −6.88028e26 9.21545e28i 3.42689e30 2.15861e29i −1.24372e32
2.20 7.55940e9i −2.98618e14 + 1.82880e15i −3.86978e19 3.78912e22i −1.38246e25 2.25737e24i 1.61394e27 1.53086e29i −3.25534e30 1.09222e30i 2.86435e32
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.65.b.a 20
3.b odd 2 1 inner 3.65.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.65.b.a 20 1.a even 1 1 trivial
3.65.b.a 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{65}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - \)\(10\!\cdots\!00\)\( T^{2} + \)\(62\!\cdots\!40\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{6} + \)\(97\!\cdots\!20\)\( T^{8} - \)\(30\!\cdots\!00\)\( T^{10} + \)\(86\!\cdots\!80\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(49\!\cdots\!60\)\( T^{16} - \)\(10\!\cdots\!00\)\( T^{18} + \)\(20\!\cdots\!48\)\( T^{20} - \)\(35\!\cdots\!00\)\( T^{22} + \)\(57\!\cdots\!60\)\( T^{24} - \)\(85\!\cdots\!00\)\( T^{26} + \)\(11\!\cdots\!80\)\( T^{28} - \)\(14\!\cdots\!00\)\( T^{30} + \)\(15\!\cdots\!20\)\( T^{32} - \)\(14\!\cdots\!00\)\( T^{34} + \)\(11\!\cdots\!40\)\( T^{36} - \)\(64\!\cdots\!00\)\( T^{38} + \)\(20\!\cdots\!76\)\( T^{40} \)
$3$ \( 1 + 1420757969558292 T + \)\(21\!\cdots\!82\)\( T^{2} + \)\(39\!\cdots\!32\)\( T^{3} - \)\(97\!\cdots\!23\)\( T^{4} - \)\(43\!\cdots\!08\)\( T^{5} - \)\(80\!\cdots\!88\)\( T^{6} - \)\(50\!\cdots\!08\)\( T^{7} + \)\(60\!\cdots\!82\)\( T^{8} + \)\(59\!\cdots\!92\)\( T^{9} + \)\(12\!\cdots\!92\)\( T^{10} + \)\(20\!\cdots\!52\)\( T^{11} + \)\(71\!\cdots\!02\)\( T^{12} - \)\(20\!\cdots\!28\)\( T^{13} - \)\(11\!\cdots\!48\)\( T^{14} - \)\(20\!\cdots\!08\)\( T^{15} - \)\(15\!\cdots\!63\)\( T^{16} + \)\(22\!\cdots\!52\)\( T^{17} + \)\(41\!\cdots\!62\)\( T^{18} + \)\(94\!\cdots\!32\)\( T^{19} + \)\(22\!\cdots\!01\)\( T^{20} \)
$5$ \( 1 - \)\(47\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!50\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{6} + \)\(19\!\cdots\!25\)\( T^{8} - \)\(17\!\cdots\!00\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{12} - \)\(77\!\cdots\!00\)\( T^{14} + \)\(43\!\cdots\!50\)\( T^{16} - \)\(23\!\cdots\!00\)\( T^{18} + \)\(12\!\cdots\!00\)\( T^{20} - \)\(67\!\cdots\!00\)\( T^{22} + \)\(37\!\cdots\!50\)\( T^{24} - \)\(19\!\cdots\!00\)\( T^{26} + \)\(92\!\cdots\!00\)\( T^{28} - \)\(37\!\cdots\!00\)\( T^{30} + \)\(12\!\cdots\!25\)\( T^{32} - \)\(32\!\cdots\!00\)\( T^{34} + \)\(61\!\cdots\!50\)\( T^{36} - \)\(77\!\cdots\!00\)\( T^{38} + \)\(48\!\cdots\!25\)\( T^{40} \)
$7$ \( ( 1 - \)\(34\!\cdots\!88\)\( T + \)\(56\!\cdots\!22\)\( T^{2} - \)\(20\!\cdots\!48\)\( T^{3} + \)\(18\!\cdots\!77\)\( T^{4} - \)\(67\!\cdots\!28\)\( T^{5} + \)\(43\!\cdots\!72\)\( T^{6} - \)\(14\!\cdots\!08\)\( T^{7} + \)\(75\!\cdots\!22\)\( T^{8} - \)\(22\!\cdots\!28\)\( T^{9} + \)\(10\!\cdots\!12\)\( T^{10} - \)\(27\!\cdots\!28\)\( T^{11} + \)\(11\!\cdots\!22\)\( T^{12} - \)\(26\!\cdots\!08\)\( T^{13} + \)\(95\!\cdots\!72\)\( T^{14} - \)\(18\!\cdots\!28\)\( T^{15} + \)\(62\!\cdots\!77\)\( T^{16} - \)\(83\!\cdots\!48\)\( T^{17} + \)\(27\!\cdots\!22\)\( T^{18} - \)\(20\!\cdots\!88\)\( T^{19} + \)\(72\!\cdots\!01\)\( T^{20} )^{2} \)
$11$ \( 1 - \)\(45\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!90\)\( T^{4} - \)\(16\!\cdots\!40\)\( T^{6} + \)\(20\!\cdots\!45\)\( T^{8} - \)\(19\!\cdots\!04\)\( T^{10} + \)\(16\!\cdots\!60\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{14} + \)\(70\!\cdots\!70\)\( T^{16} - \)\(37\!\cdots\!60\)\( T^{18} + \)\(17\!\cdots\!56\)\( T^{20} - \)\(75\!\cdots\!60\)\( T^{22} + \)\(27\!\cdots\!70\)\( T^{24} - \)\(90\!\cdots\!20\)\( T^{26} + \)\(25\!\cdots\!60\)\( T^{28} - \)\(61\!\cdots\!04\)\( T^{30} + \)\(12\!\cdots\!45\)\( T^{32} - \)\(20\!\cdots\!40\)\( T^{34} + \)\(25\!\cdots\!90\)\( T^{36} - \)\(21\!\cdots\!20\)\( T^{38} + \)\(96\!\cdots\!01\)\( T^{40} \)
$13$ \( ( 1 + \)\(37\!\cdots\!12\)\( T + \)\(11\!\cdots\!22\)\( T^{2} + \)\(18\!\cdots\!12\)\( T^{3} + \)\(55\!\cdots\!97\)\( T^{4} - \)\(33\!\cdots\!68\)\( T^{5} + \)\(17\!\cdots\!72\)\( T^{6} - \)\(18\!\cdots\!08\)\( T^{7} + \)\(47\!\cdots\!22\)\( T^{8} - \)\(65\!\cdots\!48\)\( T^{9} + \)\(10\!\cdots\!72\)\( T^{10} - \)\(12\!\cdots\!28\)\( T^{11} + \)\(18\!\cdots\!62\)\( T^{12} - \)\(14\!\cdots\!48\)\( T^{13} + \)\(26\!\cdots\!52\)\( T^{14} - \)\(98\!\cdots\!68\)\( T^{15} + \)\(31\!\cdots\!17\)\( T^{16} + \)\(20\!\cdots\!52\)\( T^{17} + \)\(24\!\cdots\!82\)\( T^{18} + \)\(16\!\cdots\!92\)\( T^{19} + \)\(83\!\cdots\!01\)\( T^{20} )^{2} \)
$17$ \( 1 - \)\(52\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!90\)\( T^{4} - \)\(22\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!45\)\( T^{8} - \)\(25\!\cdots\!00\)\( T^{10} + \)\(20\!\cdots\!80\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{14} + \)\(84\!\cdots\!10\)\( T^{16} - \)\(49\!\cdots\!00\)\( T^{18} + \)\(28\!\cdots\!48\)\( T^{20} - \)\(15\!\cdots\!00\)\( T^{22} + \)\(83\!\cdots\!10\)\( T^{24} - \)\(42\!\cdots\!00\)\( T^{26} + \)\(19\!\cdots\!80\)\( T^{28} - \)\(78\!\cdots\!00\)\( T^{30} + \)\(25\!\cdots\!45\)\( T^{32} - \)\(67\!\cdots\!00\)\( T^{34} + \)\(12\!\cdots\!90\)\( T^{36} - \)\(15\!\cdots\!00\)\( T^{38} + \)\(94\!\cdots\!01\)\( T^{40} \)
$19$ \( ( 1 - \)\(54\!\cdots\!00\)\( T + \)\(47\!\cdots\!90\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!45\)\( T^{4} - \)\(58\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} - \)\(81\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} - \)\(78\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!48\)\( T^{10} - \)\(54\!\cdots\!00\)\( T^{11} + \)\(82\!\cdots\!10\)\( T^{12} - \)\(27\!\cdots\!00\)\( T^{13} + \)\(37\!\cdots\!80\)\( T^{14} - \)\(93\!\cdots\!00\)\( T^{15} + \)\(12\!\cdots\!45\)\( T^{16} - \)\(20\!\cdots\!00\)\( T^{17} + \)\(24\!\cdots\!90\)\( T^{18} - \)\(19\!\cdots\!00\)\( T^{19} + \)\(25\!\cdots\!01\)\( T^{20} )^{2} \)
$23$ \( 1 - \)\(12\!\cdots\!00\)\( T^{2} + \)\(79\!\cdots\!90\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!45\)\( T^{8} - \)\(28\!\cdots\!00\)\( T^{10} + \)\(62\!\cdots\!80\)\( T^{12} - \)\(12\!\cdots\!00\)\( T^{14} + \)\(22\!\cdots\!10\)\( T^{16} - \)\(35\!\cdots\!00\)\( T^{18} + \)\(53\!\cdots\!48\)\( T^{20} - \)\(71\!\cdots\!00\)\( T^{22} + \)\(88\!\cdots\!10\)\( T^{24} - \)\(98\!\cdots\!00\)\( T^{26} + \)\(10\!\cdots\!80\)\( T^{28} - \)\(90\!\cdots\!00\)\( T^{30} + \)\(69\!\cdots\!45\)\( T^{32} - \)\(43\!\cdots\!00\)\( T^{34} + \)\(20\!\cdots\!90\)\( T^{36} - \)\(64\!\cdots\!00\)\( T^{38} + \)\(10\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 - \)\(39\!\cdots\!20\)\( T^{2} + \)\(78\!\cdots\!90\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(10\!\cdots\!45\)\( T^{8} - \)\(90\!\cdots\!04\)\( T^{10} + \)\(63\!\cdots\!60\)\( T^{12} - \)\(38\!\cdots\!20\)\( T^{14} + \)\(20\!\cdots\!70\)\( T^{16} - \)\(95\!\cdots\!60\)\( T^{18} + \)\(39\!\cdots\!56\)\( T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(48\!\cdots\!70\)\( T^{24} - \)\(13\!\cdots\!20\)\( T^{26} + \)\(35\!\cdots\!60\)\( T^{28} - \)\(78\!\cdots\!04\)\( T^{30} + \)\(14\!\cdots\!45\)\( T^{32} - \)\(21\!\cdots\!40\)\( T^{34} + \)\(24\!\cdots\!90\)\( T^{36} - \)\(18\!\cdots\!20\)\( T^{38} + \)\(74\!\cdots\!01\)\( T^{40} \)
$31$ \( ( 1 - \)\(22\!\cdots\!20\)\( T + \)\(13\!\cdots\!90\)\( T^{2} - \)\(20\!\cdots\!40\)\( T^{3} + \)\(86\!\cdots\!45\)\( T^{4} - \)\(30\!\cdots\!04\)\( T^{5} + \)\(34\!\cdots\!60\)\( T^{6} + \)\(43\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!70\)\( T^{8} + \)\(28\!\cdots\!40\)\( T^{9} + \)\(26\!\cdots\!56\)\( T^{10} + \)\(80\!\cdots\!40\)\( T^{11} + \)\(78\!\cdots\!70\)\( T^{12} + \)\(94\!\cdots\!80\)\( T^{13} + \)\(21\!\cdots\!60\)\( T^{14} - \)\(53\!\cdots\!04\)\( T^{15} + \)\(41\!\cdots\!45\)\( T^{16} - \)\(28\!\cdots\!40\)\( T^{17} + \)\(50\!\cdots\!90\)\( T^{18} - \)\(24\!\cdots\!20\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \)
$37$ \( ( 1 + \)\(83\!\cdots\!52\)\( T + \)\(11\!\cdots\!02\)\( T^{2} + \)\(46\!\cdots\!52\)\( T^{3} + \)\(64\!\cdots\!97\)\( T^{4} - \)\(25\!\cdots\!28\)\( T^{5} + \)\(22\!\cdots\!52\)\( T^{6} - \)\(10\!\cdots\!68\)\( T^{7} + \)\(61\!\cdots\!22\)\( T^{8} - \)\(45\!\cdots\!08\)\( T^{9} + \)\(14\!\cdots\!52\)\( T^{10} - \)\(10\!\cdots\!88\)\( T^{11} + \)\(33\!\cdots\!62\)\( T^{12} - \)\(12\!\cdots\!08\)\( T^{13} + \)\(65\!\cdots\!32\)\( T^{14} - \)\(17\!\cdots\!28\)\( T^{15} + \)\(10\!\cdots\!17\)\( T^{16} + \)\(16\!\cdots\!92\)\( T^{17} + \)\(99\!\cdots\!62\)\( T^{18} + \)\(16\!\cdots\!32\)\( T^{19} + \)\(44\!\cdots\!01\)\( T^{20} )^{2} \)
$41$ \( 1 - \)\(91\!\cdots\!20\)\( T^{2} + \)\(51\!\cdots\!90\)\( T^{4} - \)\(21\!\cdots\!40\)\( T^{6} + \)\(74\!\cdots\!45\)\( T^{8} - \)\(21\!\cdots\!04\)\( T^{10} + \)\(56\!\cdots\!60\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(27\!\cdots\!70\)\( T^{16} - \)\(51\!\cdots\!60\)\( T^{18} + \)\(88\!\cdots\!56\)\( T^{20} - \)\(13\!\cdots\!60\)\( T^{22} + \)\(20\!\cdots\!70\)\( T^{24} - \)\(26\!\cdots\!20\)\( T^{26} + \)\(31\!\cdots\!60\)\( T^{28} - \)\(33\!\cdots\!04\)\( T^{30} + \)\(30\!\cdots\!45\)\( T^{32} - \)\(24\!\cdots\!40\)\( T^{34} + \)\(15\!\cdots\!90\)\( T^{36} - \)\(77\!\cdots\!20\)\( T^{38} + \)\(23\!\cdots\!01\)\( T^{40} \)
$43$ \( ( 1 - \)\(51\!\cdots\!08\)\( T + \)\(12\!\cdots\!82\)\( T^{2} - \)\(32\!\cdots\!48\)\( T^{3} + \)\(77\!\cdots\!17\)\( T^{4} + \)\(20\!\cdots\!12\)\( T^{5} + \)\(26\!\cdots\!12\)\( T^{6} + \)\(43\!\cdots\!92\)\( T^{7} + \)\(55\!\cdots\!82\)\( T^{8} + \)\(28\!\cdots\!52\)\( T^{9} + \)\(11\!\cdots\!12\)\( T^{10} + \)\(97\!\cdots\!52\)\( T^{11} + \)\(66\!\cdots\!82\)\( T^{12} + \)\(18\!\cdots\!92\)\( T^{13} + \)\(39\!\cdots\!12\)\( T^{14} + \)\(10\!\cdots\!12\)\( T^{15} + \)\(13\!\cdots\!17\)\( T^{16} - \)\(19\!\cdots\!48\)\( T^{17} + \)\(28\!\cdots\!82\)\( T^{18} - \)\(39\!\cdots\!08\)\( T^{19} + \)\(26\!\cdots\!01\)\( T^{20} )^{2} \)
$47$ \( 1 - \)\(85\!\cdots\!00\)\( T^{2} + \)\(37\!\cdots\!90\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!45\)\( T^{8} - \)\(45\!\cdots\!00\)\( T^{10} + \)\(69\!\cdots\!80\)\( T^{12} - \)\(95\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!10\)\( T^{16} - \)\(13\!\cdots\!00\)\( T^{18} + \)\(14\!\cdots\!48\)\( T^{20} - \)\(14\!\cdots\!00\)\( T^{22} + \)\(13\!\cdots\!10\)\( T^{24} - \)\(11\!\cdots\!00\)\( T^{26} + \)\(91\!\cdots\!80\)\( T^{28} - \)\(63\!\cdots\!00\)\( T^{30} + \)\(37\!\cdots\!45\)\( T^{32} - \)\(17\!\cdots\!00\)\( T^{34} + \)\(63\!\cdots\!90\)\( T^{36} - \)\(15\!\cdots\!00\)\( T^{38} + \)\(19\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 - \)\(72\!\cdots\!00\)\( T^{2} + \)\(51\!\cdots\!90\)\( T^{4} - \)\(22\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!45\)\( T^{8} - \)\(34\!\cdots\!00\)\( T^{10} + \)\(12\!\cdots\!80\)\( T^{12} - \)\(34\!\cdots\!00\)\( T^{14} + \)\(99\!\cdots\!10\)\( T^{16} - \)\(24\!\cdots\!00\)\( T^{18} + \)\(58\!\cdots\!48\)\( T^{20} - \)\(12\!\cdots\!00\)\( T^{22} + \)\(25\!\cdots\!10\)\( T^{24} - \)\(45\!\cdots\!00\)\( T^{26} + \)\(81\!\cdots\!80\)\( T^{28} - \)\(12\!\cdots\!00\)\( T^{30} + \)\(17\!\cdots\!45\)\( T^{32} - \)\(20\!\cdots\!00\)\( T^{34} + \)\(23\!\cdots\!90\)\( T^{36} - \)\(16\!\cdots\!00\)\( T^{38} + \)\(11\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - \)\(23\!\cdots\!20\)\( T^{2} + \)\(28\!\cdots\!90\)\( T^{4} - \)\(22\!\cdots\!40\)\( T^{6} + \)\(14\!\cdots\!45\)\( T^{8} - \)\(69\!\cdots\!04\)\( T^{10} + \)\(28\!\cdots\!60\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{14} + \)\(30\!\cdots\!70\)\( T^{16} - \)\(80\!\cdots\!60\)\( T^{18} + \)\(18\!\cdots\!56\)\( T^{20} - \)\(37\!\cdots\!60\)\( T^{22} + \)\(66\!\cdots\!70\)\( T^{24} - \)\(10\!\cdots\!20\)\( T^{26} + \)\(13\!\cdots\!60\)\( T^{28} - \)\(15\!\cdots\!04\)\( T^{30} + \)\(14\!\cdots\!45\)\( T^{32} - \)\(11\!\cdots\!40\)\( T^{34} + \)\(63\!\cdots\!90\)\( T^{36} - \)\(24\!\cdots\!20\)\( T^{38} + \)\(49\!\cdots\!01\)\( T^{40} \)
$61$ \( ( 1 - \)\(22\!\cdots\!20\)\( T + \)\(88\!\cdots\!90\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!45\)\( T^{4} - \)\(84\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!60\)\( T^{6} - \)\(26\!\cdots\!20\)\( T^{7} + \)\(43\!\cdots\!70\)\( T^{8} - \)\(60\!\cdots\!60\)\( T^{9} + \)\(89\!\cdots\!56\)\( T^{10} - \)\(10\!\cdots\!60\)\( T^{11} + \)\(14\!\cdots\!70\)\( T^{12} - \)\(15\!\cdots\!20\)\( T^{13} + \)\(18\!\cdots\!60\)\( T^{14} - \)\(17\!\cdots\!04\)\( T^{15} + \)\(17\!\cdots\!45\)\( T^{16} - \)\(13\!\cdots\!40\)\( T^{17} + \)\(10\!\cdots\!90\)\( T^{18} - \)\(50\!\cdots\!20\)\( T^{19} + \)\(40\!\cdots\!01\)\( T^{20} )^{2} \)
$67$ \( ( 1 - \)\(54\!\cdots\!08\)\( T + \)\(43\!\cdots\!82\)\( T^{2} - \)\(18\!\cdots\!48\)\( T^{3} + \)\(88\!\cdots\!17\)\( T^{4} - \)\(30\!\cdots\!28\)\( T^{5} + \)\(11\!\cdots\!32\)\( T^{6} - \)\(35\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!22\)\( T^{8} - \)\(31\!\cdots\!88\)\( T^{9} + \)\(95\!\cdots\!92\)\( T^{10} - \)\(23\!\cdots\!48\)\( T^{11} + \)\(64\!\cdots\!02\)\( T^{12} - \)\(14\!\cdots\!08\)\( T^{13} + \)\(35\!\cdots\!92\)\( T^{14} - \)\(68\!\cdots\!28\)\( T^{15} + \)\(14\!\cdots\!57\)\( T^{16} - \)\(21\!\cdots\!68\)\( T^{17} + \)\(38\!\cdots\!02\)\( T^{18} - \)\(35\!\cdots\!48\)\( T^{19} + \)\(48\!\cdots\!01\)\( T^{20} )^{2} \)
$71$ \( 1 - \)\(42\!\cdots\!20\)\( T^{2} + \)\(87\!\cdots\!90\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(10\!\cdots\!45\)\( T^{8} - \)\(70\!\cdots\!04\)\( T^{10} + \)\(37\!\cdots\!60\)\( T^{12} - \)\(16\!\cdots\!20\)\( T^{14} + \)\(60\!\cdots\!70\)\( T^{16} - \)\(19\!\cdots\!60\)\( T^{18} + \)\(60\!\cdots\!56\)\( T^{20} - \)\(17\!\cdots\!60\)\( T^{22} + \)\(50\!\cdots\!70\)\( T^{24} - \)\(12\!\cdots\!20\)\( T^{26} + \)\(26\!\cdots\!60\)\( T^{28} - \)\(45\!\cdots\!04\)\( T^{30} + \)\(59\!\cdots\!45\)\( T^{32} - \)\(60\!\cdots\!40\)\( T^{34} + \)\(42\!\cdots\!90\)\( T^{36} - \)