Properties

Label 3.67.b.b
Level $3$
Weight $67$
Character orbit 3.b
Analytic conductor $82.760$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 67 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(82.7604085389\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 906717684887249855 x^{18} + \)\(34\!\cdots\!60\)\( x^{16} + \)\(71\!\cdots\!60\)\( x^{14} + \)\(89\!\cdots\!80\)\( x^{12} + \)\(68\!\cdots\!76\)\( x^{10} + \)\(31\!\cdots\!00\)\( x^{8} + \)\(84\!\cdots\!00\)\( x^{6} + \)\(11\!\cdots\!00\)\( x^{4} + \)\(60\!\cdots\!00\)\( x^{2} + \)\(56\!\cdots\!00\)\(\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{216}\cdot 3^{291}\cdot 5^{20}\cdot 7^{10}\cdot 11^{6}\cdot 13^{4} \)
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.

\(f(q)\) \(=\) \( q +\beta_{1} q^{2} +(243548779847726 + 15513 \beta_{1} - \beta_{2}) q^{3} +(-43723635666549374440 + 141 \beta_{1} - 1519 \beta_{2} + \beta_{3}) q^{4} +(-98651 + 333501149933 \beta_{1} + 493259 \beta_{2} + 3 \beta_{3} + \beta_{4}) q^{5} +(-\)\(18\!\cdots\!70\)\( - 154836520809205 \beta_{1} - 23575093 \beta_{2} - 44166 \beta_{3} + 45 \beta_{4} - \beta_{5}) q^{6} +(\)\(63\!\cdots\!16\)\( + 807734091 \beta_{1} - 8693458123 \beta_{2} + 4109226 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6}) q^{7} +(3272193665213 - 42419403620205012804 \beta_{1} - 16361058157166 \beta_{2} - 90503341 \beta_{3} + 671787 \beta_{4} + 227 \beta_{5} + 7 \beta_{7} + \beta_{8}) q^{8} +(\)\(75\!\cdots\!72\)\( + \)\(56\!\cdots\!53\)\( \beta_{1} - 204864643581341 \beta_{2} - 16029368004 \beta_{3} - 10845989 \beta_{4} - 19747 \beta_{5} + 1112 \beta_{6} - 414 \beta_{7} - 5 \beta_{8} + \beta_{9}) q^{9} +O(q^{10})\) \( q +\beta_{1} q^{2} +(243548779847726 + 15513 \beta_{1} - \beta_{2}) q^{3} +(-43723635666549374440 + 141 \beta_{1} - 1519 \beta_{2} + \beta_{3}) q^{4} +(-98651 + 333501149933 \beta_{1} + 493259 \beta_{2} + 3 \beta_{3} + \beta_{4}) q^{5} +(-\)\(18\!\cdots\!70\)\( - 154836520809205 \beta_{1} - 23575093 \beta_{2} - 44166 \beta_{3} + 45 \beta_{4} - \beta_{5}) q^{6} +(\)\(63\!\cdots\!16\)\( + 807734091 \beta_{1} - 8693458123 \beta_{2} + 4109226 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6}) q^{7} +(3272193665213 - 42419403620205012804 \beta_{1} - 16361058157166 \beta_{2} - 90503341 \beta_{3} + 671787 \beta_{4} + 227 \beta_{5} + 7 \beta_{7} + \beta_{8}) q^{8} +(\)\(75\!\cdots\!72\)\( + \)\(56\!\cdots\!53\)\( \beta_{1} - 204864643581341 \beta_{2} - 16029368004 \beta_{3} - 10845989 \beta_{4} - 19747 \beta_{5} + 1112 \beta_{6} - 414 \beta_{7} - 5 \beta_{8} + \beta_{9}) q^{9} +(-\)\(39\!\cdots\!80\)\( - 2006354176353009 \beta_{1} + 21545263826215911 \beta_{2} - 712860668503 \beta_{3} - 21061229 \beta_{4} - 2227435 \beta_{5} - 42610 \beta_{6} + 6414 \beta_{7} + 5 \beta_{8} - \beta_{10}) q^{10} +(-24769660378890561 + \)\(81\!\cdots\!44\)\( \beta_{1} + 123848982841784823 \beta_{2} + 685201679889 \beta_{3} - 4484691838 \beta_{4} + 115140705 \beta_{5} - 29842 \beta_{6} + 47894 \beta_{7} - 2871 \beta_{8} - 78 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14}) q^{11} +(\)\(36\!\cdots\!67\)\( + \)\(24\!\cdots\!32\)\( \beta_{1} + 44281020554661386717 \beta_{2} + 534479422112123 \beta_{3} - 676198095841 \beta_{4} - 216503077 \beta_{5} - 24701951 \beta_{6} + 1835810 \beta_{7} - 91989 \beta_{8} + 3394 \beta_{9} - 61 \beta_{10} - 18 \beta_{11} + 2 \beta_{12} - 3 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17}) q^{12} +(\)\(90\!\cdots\!05\)\( + 168306987341434474 \beta_{1} - 1809557276040313878 \beta_{2} + 487056592351563 \beta_{3} + 910533351 \beta_{4} - 1056867305 \beta_{5} - 146206944 \beta_{6} - 956917 \beta_{7} - 22282 \beta_{8} + 51454 \beta_{9} + 91 \beta_{10} - 13 \beta_{11} - 26 \beta_{12} - 13 \beta_{13} - 13 \beta_{15}) q^{13} +(\)\(11\!\cdots\!66\)\( + \)\(33\!\cdots\!34\)\( \beta_{1} - \)\(56\!\cdots\!40\)\( \beta_{2} - 3101175656259103 \beta_{3} + 177737010422686 \beta_{4} - 389149947578 \beta_{5} + 96713155 \beta_{6} - 61850541 \beta_{7} - 10595111 \beta_{8} - 224556 \beta_{9} + 2917 \beta_{10} + 2871 \beta_{11} + 15 \beta_{12} + 9 \beta_{13} + 312 \beta_{14} - 25 \beta_{15} - 4 \beta_{16} + 28 \beta_{17} - \beta_{18}) q^{14} +(\)\(14\!\cdots\!73\)\( + \)\(57\!\cdots\!70\)\( \beta_{1} + \)\(45\!\cdots\!55\)\( \beta_{2} + 1191470331005851167 \beta_{3} - 209192030057892 \beta_{4} - 537684921841 \beta_{5} - 5373055708 \beta_{6} - 1783497101 \beta_{7} - 104208626 \beta_{8} - 1125947 \beta_{9} + 4627 \beta_{10} + 6655 \beta_{11} - 2737 \beta_{12} + 1005 \beta_{13} - 91 \beta_{14} - 44 \beta_{15} - 685 \beta_{16} - 203 \beta_{17} - \beta_{18} - \beta_{19}) q^{15} +(\)\(17\!\cdots\!35\)\( + \)\(53\!\cdots\!52\)\( \beta_{1} - \)\(57\!\cdots\!76\)\( \beta_{2} - 59837743460457257784 \beta_{3} + 144494037007035 \beta_{4} - 27337238790086 \beta_{5} + 162598667649 \beta_{6} - 84058466325 \beta_{7} + 81866204 \beta_{8} + 2058155 \beta_{9} - 1385528 \beta_{10} - 227556 \beta_{11} + 26976 \beta_{12} - 17257 \beta_{13} + 29517 \beta_{15} - 13575 \beta_{16}) q^{16} +(-\)\(82\!\cdots\!03\)\( + \)\(25\!\cdots\!10\)\( \beta_{1} + \)\(41\!\cdots\!37\)\( \beta_{2} + 2272966005922283479 \beta_{3} - 53267819756940843 \beta_{4} + 51171855945150 \beta_{5} - 11757628214 \beta_{6} - 138617535513 \beta_{7} - 4118387223 \beta_{8} + 93532347 \beta_{9} - 376024 \beta_{10} - 354505 \beta_{11} - 10722 \beta_{12} - 9700 \beta_{13} - 73624 \beta_{14} + 8547 \beta_{15} - 1467 \beta_{16} - 24562 \beta_{17} + 154 \beta_{18} - 54 \beta_{19}) q^{17} +(-\)\(66\!\cdots\!27\)\( + \)\(18\!\cdots\!10\)\( \beta_{1} + \)\(14\!\cdots\!00\)\( \beta_{2} + \)\(10\!\cdots\!03\)\( \beta_{3} + 221181866557737097 \beta_{4} + 113870334956445 \beta_{5} + 1752875203963 \beta_{6} + 775074624591 \beta_{7} - 46623522768 \beta_{8} - 131877765 \beta_{9} + 47095812 \beta_{10} + 11499494 \beta_{11} + 562912 \beta_{12} + 50053 \beta_{13} + 45384 \beta_{14} - 711400 \beta_{15} + 714382 \beta_{16} - 48804 \beta_{17} - 1893 \beta_{18} - 192 \beta_{19}) q^{18} +(-\)\(10\!\cdots\!36\)\( + \)\(22\!\cdots\!75\)\( \beta_{1} - \)\(24\!\cdots\!76\)\( \beta_{2} - \)\(12\!\cdots\!25\)\( \beta_{3} + 27044631308377545 \beta_{4} + 3182314905096243 \beta_{5} + 28926411853036 \beta_{6} - 7859576466609 \beta_{7} - 10975544351 \beta_{8} + 8582057348 \beta_{9} + 141401103 \beta_{10} + 19863444 \beta_{11} + 1447261 \beta_{12} - 4655558 \beta_{13} - 3905707 \beta_{15} + 4253124 \beta_{16}) q^{19} +(-\)\(61\!\cdots\!27\)\( + \)\(45\!\cdots\!98\)\( \beta_{1} + \)\(30\!\cdots\!83\)\( \beta_{2} + \)\(16\!\cdots\!42\)\( \beta_{3} - 23988241282028380966 \beta_{4} + 40339083785237983 \beta_{5} - 9084273075325 \beta_{6} - 98703458731981 \beta_{7} - 955146182871 \beta_{8} + 51372893859 \beta_{9} - 282268265 \beta_{10} - 270082284 \beta_{11} - 6907347 \beta_{12} - 6646019 \beta_{13} - 20453922 \beta_{14} + 4162766 \beta_{15} - 1527457 \beta_{16} - 16366006 \beta_{17} + 74368 \beta_{18} - 6912 \beta_{19}) q^{20} +(\)\(42\!\cdots\!02\)\( + \)\(15\!\cdots\!73\)\( \beta_{1} - \)\(62\!\cdots\!34\)\( \beta_{2} - \)\(17\!\cdots\!94\)\( \beta_{3} + 19438807904075676293 \beta_{4} + 20389866953639864 \beta_{5} + 2237871417454348 \beta_{6} - 988007336580931 \beta_{7} - 6961085268831 \beta_{8} - 346939732130 \beta_{9} - 23881348 \beta_{10} - 357948321 \beta_{11} + 10406435 \beta_{12} - 1418493 \beta_{13} - 439422240 \beta_{14} + 114613126 \beta_{15} - 29955634 \beta_{16} - 36965614 \beta_{17} - 257562 \beta_{18} + 10710 \beta_{19}) q^{21} +(-\)\(96\!\cdots\!79\)\( + \)\(12\!\cdots\!78\)\( \beta_{1} - \)\(13\!\cdots\!86\)\( \beta_{2} + \)\(18\!\cdots\!15\)\( \beta_{3} + 13505675640979296233 \beta_{4} + 937188583246912087 \beta_{5} + 7317100870984344 \beta_{6} - 4527124844632888 \beta_{7} - 1580498234247 \beta_{8} - 495323279011 \beta_{9} + 7302193857 \beta_{10} + 5109872495 \beta_{11} - 2646055426 \beta_{12} - 122565280 \beta_{13} - 752214578 \beta_{15} + 287360604 \beta_{16}) q^{22} +(-\)\(86\!\cdots\!22\)\( + \)\(39\!\cdots\!46\)\( \beta_{1} + \)\(43\!\cdots\!49\)\( \beta_{2} + \)\(23\!\cdots\!46\)\( \beta_{3} - \)\(59\!\cdots\!49\)\( \beta_{4} + 696744462125720731 \beta_{5} + 8499378721672 \beta_{6} - 17534752714713524 \beta_{7} - 156975098366779 \beta_{8} + 6279328349154 \beta_{9} + 31078113 \beta_{10} + 1398970352 \beta_{11} - 890693667 \beta_{12} - 915577136 \beta_{13} + 10923005600 \beta_{14} + 370855595 \beta_{15} - 275832862 \beta_{16} - 2193895018 \beta_{17} - 3519422 \beta_{18} + 1006722 \beta_{19}) q^{23} +(-\)\(42\!\cdots\!16\)\( + \)\(39\!\cdots\!52\)\( \beta_{1} - \)\(31\!\cdots\!02\)\( \beta_{2} + \)\(10\!\cdots\!43\)\( \beta_{3} - \)\(74\!\cdots\!22\)\( \beta_{4} + 40697612525129294877 \beta_{5} + 99640010197410433 \beta_{6} - 52248700680706066 \beta_{7} - 359643966549163 \beta_{8} + 5176689414059 \beta_{9} + 1357786141460 \beta_{10} - 114231981172 \beta_{11} - 17630357444 \beta_{12} - 1489323921 \beta_{13} - 106033951520 \beta_{14} - 7215855823 \beta_{15} + 6677959609 \beta_{16} - 4948122208 \beta_{17} + 24988096 \beta_{18} + 213760 \beta_{19}) q^{24} +(-\)\(15\!\cdots\!40\)\( + \)\(19\!\cdots\!90\)\( \beta_{1} - \)\(21\!\cdots\!14\)\( \beta_{2} - \)\(37\!\cdots\!07\)\( \beta_{3} + \)\(27\!\cdots\!68\)\( \beta_{4} + 9931640817449648345 \beta_{5} + 267676016813301518 \beta_{6} - 100758455768033375 \beta_{7} + 38237234491738 \beta_{8} - 102062127425331 \beta_{9} - 5190729384525 \beta_{10} + 123555354403 \beta_{11} + 18677248831 \beta_{12} - 12361445862 \beta_{13} + 13164679142 \beta_{15} + 23682888249 \beta_{16}) q^{25} +(-\)\(24\!\cdots\!54\)\( + \)\(86\!\cdots\!34\)\( \beta_{1} + \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(66\!\cdots\!72\)\( \beta_{3} - \)\(16\!\cdots\!34\)\( \beta_{4} - \)\(18\!\cdots\!22\)\( \beta_{5} + 48141835039897638 \beta_{6} - 213749193099444522 \beta_{7} - 8823796161416340 \beta_{8} + 562068909719070 \beta_{9} + 1503903671868 \beta_{10} + 1580452796000 \beta_{11} - 7868469882 \beta_{12} + 7272229770 \beta_{13} + 1142425308752 \beta_{14} + 55309017062 \beta_{15} + 19566488552 \beta_{16} + 970071544 \beta_{17} - 15637258 \beta_{18} - 54587520 \beta_{19}) q^{26} +(\)\(27\!\cdots\!49\)\( + \)\(16\!\cdots\!77\)\( \beta_{1} - \)\(52\!\cdots\!47\)\( \beta_{2} - \)\(28\!\cdots\!15\)\( \beta_{3} + \)\(21\!\cdots\!09\)\( \beta_{4} - \)\(51\!\cdots\!68\)\( \beta_{5} - 1476677729408076276 \beta_{6} + 1302654428383585518 \beta_{7} + 9814650205801446 \beta_{8} + 485284007792700 \beta_{9} + 87788122632504 \beta_{10} - 764592440799 \beta_{11} + 367495890771 \beta_{12} - 98242695312 \beta_{13} - 1778382274383 \beta_{14} + 447710911305 \beta_{15} - 100536968922 \beta_{16} + 156262477518 \beta_{17} - 876918294 \beta_{18} - 36503766 \beta_{19}) q^{27} +(\)\(72\!\cdots\!85\)\( - \)\(44\!\cdots\!22\)\( \beta_{1} + \)\(48\!\cdots\!87\)\( \beta_{2} + \)\(16\!\cdots\!92\)\( \beta_{3} - \)\(51\!\cdots\!72\)\( \beta_{4} - \)\(24\!\cdots\!01\)\( \beta_{5} - 65536019662454142629 \beta_{6} + 18400113042966923233 \beta_{7} + 5893954219022009 \beta_{8} - 5000185140655103 \beta_{9} - 378394054518613 \beta_{10} - 4685479887872 \beta_{11} + 1734873544607 \beta_{12} + 1179135840603 \beta_{13} + 1876365803536 \beta_{15} - 723854031789 \beta_{16}) q^{28} +(\)\(61\!\cdots\!57\)\( + \)\(42\!\cdots\!63\)\( \beta_{1} - \)\(30\!\cdots\!71\)\( \beta_{2} - \)\(16\!\cdots\!09\)\( \beta_{3} + \)\(59\!\cdots\!95\)\( \beta_{4} - \)\(12\!\cdots\!42\)\( \beta_{5} + 121666834561016212 \beta_{6} + 11945253476111186986 \beta_{7} - 449980534057236588 \beta_{8} + 30705636196064898 \beta_{9} + 580396076338 \beta_{10} + 4895998772150 \beta_{11} + 719866807878 \beta_{12} + 2040348825720 \beta_{13} - 9234470847856 \beta_{14} + 4080840349252 \beta_{15} + 2076982725862 \beta_{16} + 3494677340504 \beta_{17} + 5542446472 \beta_{18} + 1794069000 \beta_{19}) q^{29} +(-\)\(67\!\cdots\!34\)\( - \)\(70\!\cdots\!12\)\( \beta_{1} + \)\(29\!\cdots\!88\)\( \beta_{2} + \)\(15\!\cdots\!21\)\( \beta_{3} - \)\(25\!\cdots\!14\)\( \beta_{4} - \)\(96\!\cdots\!46\)\( \beta_{5} - \)\(11\!\cdots\!69\)\( \beta_{6} + 54033481224259953607 \beta_{7} + 2875208016481719003 \beta_{8} + 10101868009202120 \beta_{9} + 1715894895749635 \beta_{10} + 150440720387079 \beta_{11} + 2899847729401 \beta_{12} + 12554926265769 \beta_{13} + 100617707580024 \beta_{14} + 15455817910817 \beta_{15} + 6340391089552 \beta_{16} + 1292468245852 \beta_{17} + 15355393119 \beta_{18} + 1657384704 \beta_{19}) q^{30} +(\)\(30\!\cdots\!10\)\( + \)\(73\!\cdots\!75\)\( \beta_{1} - \)\(78\!\cdots\!64\)\( \beta_{2} - \)\(21\!\cdots\!14\)\( \beta_{3} + \)\(11\!\cdots\!94\)\( \beta_{4} + \)\(22\!\cdots\!79\)\( \beta_{5} - 90374444641799555087 \beta_{6} - 25793630245299129516 \beta_{7} + 49693062020770031 \beta_{8} - 229070481196549496 \beta_{9} + 437251064697033 \beta_{10} + 186711635265300 \beta_{11} - 43300939660921 \beta_{12} - 20742683333014 \beta_{13} + 32825735136079 \beta_{15} + 26712084480864 \beta_{16}) q^{31} +(-\)\(58\!\cdots\!84\)\( + \)\(29\!\cdots\!08\)\( \beta_{1} + \)\(29\!\cdots\!72\)\( \beta_{2} + \)\(16\!\cdots\!84\)\( \beta_{3} - \)\(78\!\cdots\!80\)\( \beta_{4} + \)\(53\!\cdots\!16\)\( \beta_{5} - 7565690328956741408 \beta_{6} - \)\(11\!\cdots\!68\)\( \beta_{7} - 59900743195816302480 \beta_{8} + 1022882632440221664 \beta_{9} - 481859934597664 \beta_{10} - 284903221474432 \beta_{11} - 52705914990816 \beta_{12} - 26555106345184 \beta_{13} - 806727011023936 \beta_{14} + 115585109859648 \beta_{15} + 23252169322464 \beta_{16} - 93242037165760 \beta_{17} - 216768355712 \beta_{18} - 41204602368 \beta_{19}) q^{32} +(\)\(38\!\cdots\!20\)\( - \)\(35\!\cdots\!47\)\( \beta_{1} - \)\(17\!\cdots\!44\)\( \beta_{2} - \)\(35\!\cdots\!35\)\( \beta_{3} - \)\(10\!\cdots\!66\)\( \beta_{4} + \)\(43\!\cdots\!33\)\( \beta_{5} - \)\(47\!\cdots\!22\)\( \beta_{6} - \)\(96\!\cdots\!53\)\( \beta_{7} + 92593382109836726144 \beta_{8} + 808231311904196486 \beta_{9} - 53419517891892424 \beta_{10} + 393453796633061 \beta_{11} - 326818675494404 \beta_{12} - 278204770840398 \beta_{13} - 218754111114872 \beta_{14} + 144236512488095 \beta_{15} + 117522174876481 \beta_{16} - 121905928776550 \beta_{17} - 78046105634 \beta_{18} - 46270605266 \beta_{19}) q^{33} +(-\)\(30\!\cdots\!96\)\( + \)\(70\!\cdots\!96\)\( \beta_{1} - \)\(75\!\cdots\!18\)\( \beta_{2} + \)\(68\!\cdots\!96\)\( \beta_{3} + \)\(68\!\cdots\!62\)\( \beta_{4} + \)\(13\!\cdots\!98\)\( \beta_{5} + \)\(23\!\cdots\!64\)\( \beta_{6} - \)\(15\!\cdots\!00\)\( \beta_{7} - 2037296753024088378 \beta_{8} - 2371754963378464784 \beta_{9} + 181383093791780274 \beta_{10} + 7108370736424204 \beta_{11} - 236772159255402 \beta_{12} + 356438552128928 \beta_{13} - 25729765626138 \beta_{15} + 383399006202444 \beta_{16}) q^{34} +(\)\(15\!\cdots\!09\)\( + \)\(22\!\cdots\!61\)\( \beta_{1} - \)\(79\!\cdots\!86\)\( \beta_{2} - \)\(43\!\cdots\!68\)\( \beta_{3} - \)\(42\!\cdots\!36\)\( \beta_{4} - \)\(20\!\cdots\!83\)\( \beta_{5} + \)\(42\!\cdots\!90\)\( \beta_{6} + \)\(15\!\cdots\!81\)\( \beta_{7} - \)\(55\!\cdots\!69\)\( \beta_{8} - 4297672365199886754 \beta_{9} + 15484512503170405 \beta_{10} + 14404370497801199 \beta_{11} + 375997738698462 \beta_{12} + 270458143511904 \beta_{13} + 16559246384968367 \beta_{14} - 561215207596046 \beta_{15} - 52239882653108 \beta_{16} + 773715794484836 \beta_{17} + 4885545373612 \beta_{18} + 705065200812 \beta_{19}) q^{35} +(-\)\(15\!\cdots\!63\)\( - \)\(99\!\cdots\!57\)\( \beta_{1} - \)\(43\!\cdots\!68\)\( \beta_{2} + \)\(43\!\cdots\!63\)\( \beta_{3} + \)\(37\!\cdots\!54\)\( \beta_{4} - \)\(16\!\cdots\!49\)\( \beta_{5} - \)\(16\!\cdots\!85\)\( \beta_{6} + \)\(52\!\cdots\!15\)\( \beta_{7} + \)\(20\!\cdots\!89\)\( \beta_{8} - 14885539198748789237 \beta_{9} - 222294880441696869 \beta_{10} + 14077789995463596 \beta_{11} + 3950896511557197 \beta_{12} + 4213898784405837 \beta_{13} - 28187902605836142 \beta_{14} - 3406239189593766 \beta_{15} - 2240276838825861 \beta_{16} + 2312711848419462 \beta_{17} - 2951491215744 \beta_{18} + 929160495360 \beta_{19}) q^{36} +(\)\(13\!\cdots\!55\)\( - \)\(10\!\cdots\!50\)\( \beta_{1} + \)\(10\!\cdots\!58\)\( \beta_{2} - \)\(58\!\cdots\!83\)\( \beta_{3} - \)\(10\!\cdots\!69\)\( \beta_{4} - \)\(74\!\cdots\!51\)\( \beta_{5} + \)\(26\!\cdots\!48\)\( \beta_{6} + \)\(33\!\cdots\!65\)\( \beta_{7} - 24063364435328543586 \beta_{8} + 93381905261382060112 \beta_{9} - 1502896701506904547 \beta_{10} - 36885515415962599 \beta_{11} + 16446456371904672 \beta_{12} - 3420748957243449 \beta_{13} - 2495248454035269 \beta_{15} - 15535116457689030 \beta_{16}) q^{37} +(\)\(70\!\cdots\!51\)\( - \)\(31\!\cdots\!20\)\( \beta_{1} - \)\(35\!\cdots\!18\)\( \beta_{2} - \)\(19\!\cdots\!45\)\( \beta_{3} + \)\(19\!\cdots\!47\)\( \beta_{4} - \)\(83\!\cdots\!07\)\( \beta_{5} + \)\(19\!\cdots\!98\)\( \beta_{6} + \)\(78\!\cdots\!94\)\( \beta_{7} - \)\(28\!\cdots\!65\)\( \beta_{8} - \)\(26\!\cdots\!09\)\( \beta_{9} + 604479537347376095 \beta_{10} + 560341698491702539 \beta_{11} + 7398083809725978 \beta_{12} - 322530960064522 \beta_{13} - 30115526084926256 \beta_{14} - 29574679965329302 \beta_{15} - 8981198160985432 \beta_{16} + 7686685667906856 \beta_{17} - 75560632793478 \beta_{18} - 9253218719232 \beta_{19}) q^{38} +(\)\(27\!\cdots\!71\)\( - \)\(18\!\cdots\!85\)\( \beta_{1} - \)\(90\!\cdots\!69\)\( \beta_{2} - \)\(76\!\cdots\!45\)\( \beta_{3} - \)\(12\!\cdots\!32\)\( \beta_{4} + \)\(60\!\cdots\!32\)\( \beta_{5} - \)\(95\!\cdots\!23\)\( \beta_{6} - \)\(45\!\cdots\!73\)\( \beta_{7} - \)\(18\!\cdots\!07\)\( \beta_{8} + 15658614117589236391 \beta_{9} + 13684085684161320350 \beta_{10} + 1101929231445107559 \beta_{11} + 31197307485333938 \beta_{12} - 45882753554248143 \beta_{13} + 386085803024044149 \beta_{14} - 20720496663237683 \beta_{15} - 11117916377855755 \beta_{16} - 19114388289666397 \beta_{17} + 92459741647689 \beta_{18} - 14314753327095 \beta_{19}) q^{39} +(-\)\(82\!\cdots\!70\)\( + \)\(34\!\cdots\!32\)\( \beta_{1} - \)\(37\!\cdots\!44\)\( \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3} + \)\(37\!\cdots\!54\)\( \beta_{4} + \)\(70\!\cdots\!60\)\( \beta_{5} + \)\(85\!\cdots\!02\)\( \beta_{6} - \)\(84\!\cdots\!62\)\( \beta_{7} - \)\(15\!\cdots\!88\)\( \beta_{8} - \)\(10\!\cdots\!74\)\( \beta_{9} - 8499369835688002792 \beta_{10} + 5270851446735021512 \beta_{11} - 195488930021800376 \beta_{12} - 8320436061065798 \beta_{13} - 183773021170782682 \beta_{15} + 159832009611998646 \beta_{16}) q^{40} +(-\)\(26\!\cdots\!04\)\( - \)\(41\!\cdots\!86\)\( \beta_{1} + \)\(13\!\cdots\!11\)\( \beta_{2} + \)\(72\!\cdots\!90\)\( \beta_{3} - \)\(17\!\cdots\!37\)\( \beta_{4} - \)\(45\!\cdots\!35\)\( \beta_{5} + \)\(13\!\cdots\!44\)\( \beta_{6} - \)\(99\!\cdots\!06\)\( \beta_{7} + \)\(82\!\cdots\!67\)\( \beta_{8} + \)\(13\!\cdots\!26\)\( \beta_{9} + 3423431548333403395 \beta_{10} + 3734462031573249496 \beta_{11} - 121378894588318257 \beta_{12} - 93847377761042256 \beta_{13} - 2608687009127923208 \beta_{14} + 151437049866236937 \beta_{15} + 5960235914252598 \beta_{16} - 257477902390076718 \beta_{17} + 841844375306886 \beta_{18} + 93374709914070 \beta_{19}) q^{41} +(-\)\(18\!\cdots\!28\)\( + \)\(12\!\cdots\!39\)\( \beta_{1} + \)\(61\!\cdots\!59\)\( \beta_{2} + \)\(86\!\cdots\!07\)\( \beta_{3} + \)\(41\!\cdots\!39\)\( \beta_{4} - \)\(40\!\cdots\!71\)\( \beta_{5} - \)\(27\!\cdots\!94\)\( \beta_{6} - \)\(13\!\cdots\!34\)\( \beta_{7} - \)\(30\!\cdots\!43\)\( \beta_{8} + \)\(73\!\cdots\!76\)\( \beta_{9} - 13316069706343073545 \beta_{10} + 33921596468841078764 \beta_{11} - 1332397914148435298 \beta_{12} + 199329476970357588 \beta_{13} - 215577088867238432 \beta_{14} - 67089424568889298 \beta_{15} + 295166704849728052 \beta_{16} - 2187488270765392 \beta_{17} - 1502585694337172 \beta_{18} + 174549301330432 \beta_{19}) q^{42} +(\)\(14\!\cdots\!28\)\( - \)\(16\!\cdots\!61\)\( \beta_{1} + \)\(17\!\cdots\!44\)\( \beta_{2} + \)\(18\!\cdots\!27\)\( \beta_{3} - \)\(77\!\cdots\!87\)\( \beta_{4} + \)\(68\!\cdots\!07\)\( \beta_{5} - \)\(11\!\cdots\!20\)\( \beta_{6} + \)\(67\!\cdots\!03\)\( \beta_{7} - \)\(14\!\cdots\!71\)\( \beta_{8} - \)\(77\!\cdots\!08\)\( \beta_{9} + \)\(25\!\cdots\!15\)\( \beta_{10} + 56668649224536541412 \beta_{11} + 592196811077017277 \beta_{12} + 376646756131971362 \beta_{13} + 3020831181832291381 \beta_{15} - 989420785105561092 \beta_{16}) q^{43} +(\)\(20\!\cdots\!17\)\( - \)\(18\!\cdots\!98\)\( \beta_{1} - \)\(10\!\cdots\!97\)\( \beta_{2} - \)\(56\!\cdots\!46\)\( \beta_{3} + \)\(30\!\cdots\!74\)\( \beta_{4} - \)\(31\!\cdots\!05\)\( \beta_{5} + \)\(79\!\cdots\!39\)\( \beta_{6} + \)\(15\!\cdots\!79\)\( \beta_{7} + \)\(75\!\cdots\!17\)\( \beta_{8} + \)\(19\!\cdots\!87\)\( \beta_{9} + \)\(23\!\cdots\!47\)\( \beta_{10} + \)\(23\!\cdots\!84\)\( \beta_{11} + 734577223072904121 \beta_{12} + 1534408937685069513 \beta_{13} + 36857148901415351782 \beta_{14} + 2354484349189565974 \beta_{15} + 1345763797775248291 \beta_{16} + 2836712396227838114 \beta_{17} - 6631768701337088 \beta_{18} - 698446755210240 \beta_{19}) q^{44} +(\)\(14\!\cdots\!17\)\( + \)\(29\!\cdots\!25\)\( \beta_{1} - \)\(11\!\cdots\!92\)\( \beta_{2} - \)\(55\!\cdots\!45\)\( \beta_{3} + \)\(39\!\cdots\!62\)\( \beta_{4} - \)\(46\!\cdots\!31\)\( \beta_{5} - \)\(13\!\cdots\!72\)\( \beta_{6} + \)\(36\!\cdots\!12\)\( \beta_{7} - \)\(62\!\cdots\!15\)\( \beta_{8} - \)\(21\!\cdots\!74\)\( \beta_{9} - \)\(87\!\cdots\!15\)\( \beta_{10} + \)\(24\!\cdots\!86\)\( \beta_{11} + 18203294087304472385 \beta_{12} + 1505755454302869716 \beta_{13} - 40245516658730326416 \beta_{14} + 11376971343588290605 \beta_{15} - 1027867524435997192 \beta_{16} + 1696702787237689782 \beta_{17} + 16768080674269554 \beta_{18} - 1704123345500286 \beta_{19}) q^{45} +(-\)\(46\!\cdots\!86\)\( + \)\(96\!\cdots\!40\)\( \beta_{1} - \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(23\!\cdots\!66\)\( \beta_{3} + \)\(87\!\cdots\!94\)\( \beta_{4} + \)\(92\!\cdots\!82\)\( \beta_{5} - \)\(34\!\cdots\!16\)\( \beta_{6} + \)\(53\!\cdots\!20\)\( \beta_{7} - \)\(26\!\cdots\!90\)\( \beta_{8} + \)\(27\!\cdots\!02\)\( \beta_{9} + \)\(13\!\cdots\!86\)\( \beta_{10} + \)\(77\!\cdots\!14\)\( \beta_{11} + 11614594066073560800 \beta_{12} - 2182980601754876672 \beta_{13} - 11892437666856537888 \beta_{15} + 8106690016046478528 \beta_{16}) q^{46} +(\)\(18\!\cdots\!82\)\( + \)\(10\!\cdots\!50\)\( \beta_{1} - \)\(92\!\cdots\!66\)\( \beta_{2} - \)\(52\!\cdots\!92\)\( \beta_{3} - \)\(31\!\cdots\!34\)\( \beta_{4} - \)\(13\!\cdots\!00\)\( \beta_{5} + \)\(34\!\cdots\!16\)\( \beta_{6} - \)\(87\!\cdots\!14\)\( \beta_{7} - \)\(72\!\cdots\!92\)\( \beta_{8} - \)\(12\!\cdots\!08\)\( \beta_{9} + \)\(10\!\cdots\!60\)\( \beta_{10} + \)\(99\!\cdots\!98\)\( \beta_{11} - 3058790064395549874 \beta_{12} - 7641996230060385504 \beta_{13} - \)\(17\!\cdots\!22\)\( \beta_{14} - 14049308196535584494 \beta_{15} - 7395991619282240324 \beta_{16} - 13468563642446704348 \beta_{17} + 31338547099043884 \beta_{18} + 3296731963707756 \beta_{19}) q^{47} +(\)\(22\!\cdots\!59\)\( - \)\(99\!\cdots\!60\)\( \beta_{1} - \)\(18\!\cdots\!48\)\( \beta_{2} + \)\(61\!\cdots\!64\)\( \beta_{3} + \)\(12\!\cdots\!23\)\( \beta_{4} + \)\(21\!\cdots\!22\)\( \beta_{5} + \)\(28\!\cdots\!25\)\( \beta_{6} - \)\(28\!\cdots\!89\)\( \beta_{7} + \)\(18\!\cdots\!64\)\( \beta_{8} - \)\(83\!\cdots\!13\)\( \beta_{9} + \)\(15\!\cdots\!36\)\( \beta_{10} + \)\(32\!\cdots\!40\)\( \beta_{11} - \)\(15\!\cdots\!96\)\( \beta_{12} - 20557299137900844525 \beta_{13} + \)\(41\!\cdots\!92\)\( \beta_{14} - \)\(11\!\cdots\!67\)\( \beta_{15} + 2013731952482154429 \beta_{16} - 17175823382974874304 \beta_{17} - 135278111441676672 \beta_{18} + 13212709947873792 \beta_{19}) q^{48} +(\)\(17\!\cdots\!40\)\( + \)\(34\!\cdots\!14\)\( \beta_{1} - \)\(37\!\cdots\!74\)\( \beta_{2} - \)\(17\!\cdots\!45\)\( \beta_{3} + \)\(41\!\cdots\!20\)\( \beta_{4} + \)\(44\!\cdots\!99\)\( \beta_{5} + \)\(12\!\cdots\!42\)\( \beta_{6} - \)\(12\!\cdots\!49\)\( \beta_{7} - \)\(10\!\cdots\!50\)\( \beta_{8} + \)\(17\!\cdots\!03\)\( \beta_{9} - \)\(17\!\cdots\!39\)\( \beta_{10} + \)\(34\!\cdots\!85\)\( \beta_{11} - \)\(20\!\cdots\!15\)\( \beta_{12} + 6361926049382228794 \beta_{13} - 26382667440693627714 \beta_{15} - 70650975683798855541 \beta_{16}) q^{49} +(\)\(10\!\cdots\!50\)\( + \)\(10\!\cdots\!25\)\( \beta_{1} - \)\(54\!\cdots\!00\)\( \beta_{2} - \)\(31\!\cdots\!00\)\( \beta_{3} - \)\(40\!\cdots\!50\)\( \beta_{4} - \)\(52\!\cdots\!50\)\( \beta_{5} + \)\(12\!\cdots\!50\)\( \beta_{6} - \)\(30\!\cdots\!50\)\( \beta_{7} - \)\(52\!\cdots\!00\)\( \beta_{8} - \)\(10\!\cdots\!50\)\( \beta_{9} + \)\(37\!\cdots\!00\)\( \beta_{10} + \)\(36\!\cdots\!00\)\( \beta_{11} + 4083867489089985150 \beta_{12} - 32295314607163363950 \beta_{13} - \)\(71\!\cdots\!00\)\( \beta_{14} - \)\(12\!\cdots\!50\)\( \beta_{15} - 50882821331250104600 \beta_{16} - 38241092589882317800 \beta_{17} + 14961606940315150 \beta_{18} + 108488553206400 \beta_{19}) q^{50} +(\)\(12\!\cdots\!83\)\( - \)\(19\!\cdots\!83\)\( \beta_{1} - \)\(16\!\cdots\!22\)\( \beta_{2} - \)\(42\!\cdots\!56\)\( \beta_{3} + \)\(87\!\cdots\!92\)\( \beta_{4} - \)\(50\!\cdots\!63\)\( \beta_{5} + \)\(50\!\cdots\!60\)\( \beta_{6} + \)\(15\!\cdots\!21\)\( \beta_{7} + \)\(52\!\cdots\!41\)\( \beta_{8} - \)\(28\!\cdots\!04\)\( \beta_{9} - \)\(20\!\cdots\!15\)\( \beta_{10} - \)\(26\!\cdots\!55\)\( \beta_{11} + \)\(79\!\cdots\!46\)\( \beta_{12} + 20981125375805048586 \beta_{13} - \)\(11\!\cdots\!91\)\( \beta_{14} + \)\(30\!\cdots\!20\)\( \beta_{15} - \)\(10\!\cdots\!54\)\( \beta_{16} + 57686974309737340162 \beta_{17} + 755044498853112326 \beta_{18} - 77812741139237050 \beta_{19}) q^{51} +(-\)\(35\!\cdots\!76\)\( - \)\(18\!\cdots\!10\)\( \beta_{1} + \)\(19\!\cdots\!74\)\( \beta_{2} + \)\(18\!\cdots\!34\)\( \beta_{3} - \)\(22\!\cdots\!68\)\( \beta_{4} - \)\(54\!\cdots\!40\)\( \beta_{5} - \)\(20\!\cdots\!20\)\( \beta_{6} + \)\(86\!\cdots\!60\)\( \beta_{7} - \)\(48\!\cdots\!88\)\( \beta_{8} + \)\(12\!\cdots\!56\)\( \beta_{9} + \)\(11\!\cdots\!40\)\( \beta_{10} - \)\(20\!\cdots\!64\)\( \beta_{11} + \)\(16\!\cdots\!96\)\( \beta_{12} - \)\(13\!\cdots\!64\)\( \beta_{13} - \)\(29\!\cdots\!12\)\( \beta_{15} + \)\(41\!\cdots\!84\)\( \beta_{16}) q^{52} +(\)\(31\!\cdots\!09\)\( - \)\(56\!\cdots\!71\)\( \beta_{1} - \)\(15\!\cdots\!67\)\( \beta_{2} - \)\(89\!\cdots\!89\)\( \beta_{3} - \)\(32\!\cdots\!13\)\( \beta_{4} - \)\(17\!\cdots\!30\)\( \beta_{5} + \)\(26\!\cdots\!12\)\( \beta_{6} + \)\(58\!\cdots\!92\)\( \beta_{7} + \)\(34\!\cdots\!06\)\( \beta_{8} + \)\(86\!\cdots\!84\)\( \beta_{9} + \)\(22\!\cdots\!30\)\( \beta_{10} + \)\(32\!\cdots\!36\)\( \beta_{11} + \)\(25\!\cdots\!42\)\( \beta_{12} + \)\(58\!\cdots\!52\)\( \beta_{13} + \)\(14\!\cdots\!96\)\( \beta_{14} + \)\(99\!\cdots\!82\)\( \beta_{15} + \)\(54\!\cdots\!52\)\( \beta_{16} + \)\(10\!\cdots\!44\)\( \beta_{17} - 1850377327283043572 \beta_{18} - 179313044337464148 \beta_{19}) q^{53} +(-\)\(19\!\cdots\!97\)\( + \)\(48\!\cdots\!49\)\( \beta_{1} + \)\(69\!\cdots\!05\)\( \beta_{2} - \)\(81\!\cdots\!53\)\( \beta_{3} + \)\(68\!\cdots\!02\)\( \beta_{4} - \)\(67\!\cdots\!26\)\( \beta_{5} - \)\(11\!\cdots\!42\)\( \beta_{6} + \)\(45\!\cdots\!74\)\( \beta_{7} - \)\(90\!\cdots\!41\)\( \beta_{8} + \)\(19\!\cdots\!97\)\( \beta_{9} - \)\(49\!\cdots\!45\)\( \beta_{10} - \)\(74\!\cdots\!91\)\( \beta_{11} - \)\(10\!\cdots\!60\)\( \beta_{12} + \)\(10\!\cdots\!74\)\( \beta_{13} - \)\(10\!\cdots\!72\)\( \beta_{14} + \)\(18\!\cdots\!88\)\( \beta_{15} + \)\(74\!\cdots\!00\)\( \beta_{16} + \)\(37\!\cdots\!44\)\( \beta_{17} - 2112081418711566918 \beta_{18} + 292454857626024960 \beta_{19}) q^{54} +(\)\(65\!\cdots\!70\)\( + \)\(86\!\cdots\!40\)\( \beta_{1} - \)\(92\!\cdots\!33\)\( \beta_{2} - \)\(49\!\cdots\!64\)\( \beta_{3} + \)\(12\!\cdots\!51\)\( \beta_{4} - \)\(22\!\cdots\!85\)\( \beta_{5} + \)\(17\!\cdots\!16\)\( \beta_{6} - \)\(27\!\cdots\!60\)\( \beta_{7} + \)\(79\!\cdots\!31\)\( \beta_{8} - \)\(34\!\cdots\!72\)\( \beta_{9} - \)\(76\!\cdots\!35\)\( \beta_{10} - \)\(20\!\cdots\!64\)\( \beta_{11} - \)\(79\!\cdots\!53\)\( \beta_{12} + \)\(19\!\cdots\!06\)\( \beta_{13} + \)\(92\!\cdots\!79\)\( \beta_{15} - \)\(29\!\cdots\!12\)\( \beta_{16}) q^{55} +(-\)\(43\!\cdots\!14\)\( - \)\(67\!\cdots\!56\)\( \beta_{1} + \)\(21\!\cdots\!04\)\( \beta_{2} + \)\(11\!\cdots\!74\)\( \beta_{3} - \)\(35\!\cdots\!46\)\( \beta_{4} + \)\(74\!\cdots\!22\)\( \beta_{5} - \)\(17\!\cdots\!00\)\( \beta_{6} - \)\(33\!\cdots\!74\)\( \beta_{7} + \)\(17\!\cdots\!94\)\( \beta_{8} + \)\(30\!\cdots\!20\)\( \beta_{9} - \)\(54\!\cdots\!28\)\( \beta_{10} - \)\(53\!\cdots\!12\)\( \beta_{11} - \)\(32\!\cdots\!64\)\( \beta_{12} - \)\(27\!\cdots\!08\)\( \beta_{13} - \)\(77\!\cdots\!68\)\( \beta_{14} + \)\(33\!\cdots\!44\)\( \beta_{15} - \)\(15\!\cdots\!28\)\( \beta_{16} - \)\(72\!\cdots\!48\)\( \beta_{17} + 19167543270266925696 \beta_{18} + 2005471050215109120 \beta_{19}) q^{56} +(\)\(72\!\cdots\!09\)\( + \)\(91\!\cdots\!25\)\( \beta_{1} + \)\(12\!\cdots\!74\)\( \beta_{2} - \)\(26\!\cdots\!56\)\( \beta_{3} + \)\(34\!\cdots\!62\)\( \beta_{4} + \)\(85\!\cdots\!88\)\( \beta_{5} - \)\(13\!\cdots\!32\)\( \beta_{6} - \)\(12\!\cdots\!74\)\( \beta_{7} - \)\(15\!\cdots\!86\)\( \beta_{8} + \)\(16\!\cdots\!19\)\( \beta_{9} + \)\(21\!\cdots\!07\)\( \beta_{10} - \)\(50\!\cdots\!78\)\( \beta_{11} - \)\(22\!\cdots\!49\)\( \beta_{12} - \)\(87\!\cdots\!52\)\( \beta_{13} + \)\(10\!\cdots\!40\)\( \beta_{14} - \)\(12\!\cdots\!49\)\( \beta_{15} + \)\(22\!\cdots\!64\)\( \beta_{16} - \)\(55\!\cdots\!94\)\( \beta_{17} - 9079817399649036126 \beta_{18} + 86086161768973266 \beta_{19}) q^{57} +(-\)\(50\!\cdots\!68\)\( - \)\(88\!\cdots\!43\)\( \beta_{1} + \)\(94\!\cdots\!73\)\( \beta_{2} + \)\(77\!\cdots\!19\)\( \beta_{3} - \)\(22\!\cdots\!31\)\( \beta_{4} - \)\(80\!\cdots\!65\)\( \beta_{5} + \)\(47\!\cdots\!10\)\( \beta_{6} + \)\(14\!\cdots\!98\)\( \beta_{7} + \)\(14\!\cdots\!83\)\( \beta_{8} + \)\(15\!\cdots\!04\)\( \beta_{9} - \)\(38\!\cdots\!43\)\( \beta_{10} - \)\(64\!\cdots\!60\)\( \beta_{11} + \)\(98\!\cdots\!44\)\( \beta_{12} - \)\(12\!\cdots\!60\)\( \beta_{13} - \)\(66\!\cdots\!08\)\( \beta_{15} + \)\(34\!\cdots\!44\)\( \beta_{16}) q^{58} +(\)\(35\!\cdots\!78\)\( + \)\(19\!\cdots\!77\)\( \beta_{1} - \)\(17\!\cdots\!75\)\( \beta_{2} - \)\(99\!\cdots\!89\)\( \beta_{3} - \)\(74\!\cdots\!32\)\( \beta_{4} + \)\(87\!\cdots\!54\)\( \beta_{5} - \)\(24\!\cdots\!36\)\( \beta_{6} + \)\(13\!\cdots\!39\)\( \beta_{7} - \)\(83\!\cdots\!20\)\( \beta_{8} - \)\(42\!\cdots\!80\)\( \beta_{9} - \)\(66\!\cdots\!96\)\( \beta_{10} - \)\(74\!\cdots\!60\)\( \beta_{11} + \)\(14\!\cdots\!64\)\( \beta_{12} + \)\(22\!\cdots\!00\)\( \beta_{13} + \)\(27\!\cdots\!76\)\( \beta_{14} - \)\(47\!\cdots\!44\)\( \beta_{15} - \)\(13\!\cdots\!04\)\( \beta_{16} + \)\(18\!\cdots\!52\)\( \beta_{17} - \)\(11\!\cdots\!44\)\( \beta_{18} - 13482718801134233400 \beta_{19}) q^{59} +(\)\(94\!\cdots\!47\)\( - \)\(13\!\cdots\!78\)\( \beta_{1} - \)\(74\!\cdots\!31\)\( \beta_{2} - \)\(24\!\cdots\!22\)\( \beta_{3} + \)\(11\!\cdots\!90\)\( \beta_{4} + \)\(25\!\cdots\!81\)\( \beta_{5} + \)\(13\!\cdots\!69\)\( \beta_{6} + \)\(63\!\cdots\!09\)\( \beta_{7} + \)\(31\!\cdots\!67\)\( \beta_{8} - \)\(19\!\cdots\!95\)\( \beta_{9} - \)\(90\!\cdots\!59\)\( \beta_{10} - \)\(18\!\cdots\!64\)\( \beta_{11} + \)\(21\!\cdots\!59\)\( \beta_{12} + \)\(27\!\cdots\!31\)\( \beta_{13} - \)\(29\!\cdots\!14\)\( \beta_{14} + \)\(17\!\cdots\!98\)\( \beta_{15} - \)\(47\!\cdots\!87\)\( \beta_{16} + \)\(26\!\cdots\!38\)\( \beta_{17} + \)\(16\!\cdots\!96\)\( \beta_{18} - 12220617046878639104 \beta_{19}) q^{60} +(\)\(18\!\cdots\!55\)\( - \)\(73\!\cdots\!66\)\( \beta_{1} + \)\(78\!\cdots\!74\)\( \beta_{2} + \)\(14\!\cdots\!65\)\( \beta_{3} - \)\(10\!\cdots\!41\)\( \beta_{4} - \)\(25\!\cdots\!03\)\( \beta_{5} - \)\(44\!\cdots\!36\)\( \beta_{6} + \)\(19\!\cdots\!85\)\( \beta_{7} + \)\(43\!\cdots\!90\)\( \beta_{8} + \)\(36\!\cdots\!92\)\( \beta_{9} - \)\(31\!\cdots\!39\)\( \beta_{10} - \)\(18\!\cdots\!11\)\( \beta_{11} + \)\(11\!\cdots\!40\)\( \beta_{12} + \)\(26\!\cdots\!03\)\( \beta_{13} + \)\(22\!\cdots\!07\)\( \beta_{15} - \)\(28\!\cdots\!82\)\( \beta_{16}) q^{61} +(\)\(44\!\cdots\!40\)\( + \)\(18\!\cdots\!14\)\( \beta_{1} - \)\(22\!\cdots\!72\)\( \beta_{2} - \)\(12\!\cdots\!55\)\( \beta_{3} + \)\(22\!\cdots\!44\)\( \beta_{4} - \)\(81\!\cdots\!32\)\( \beta_{5} + \)\(19\!\cdots\!49\)\( \beta_{6} + \)\(22\!\cdots\!09\)\( \beta_{7} - \)\(38\!\cdots\!11\)\( \beta_{8} + \)\(34\!\cdots\!38\)\( \beta_{9} + \)\(60\!\cdots\!37\)\( \beta_{10} + \)\(60\!\cdots\!91\)\( \beta_{11} + \)\(26\!\cdots\!53\)\( \beta_{12} + \)\(40\!\cdots\!67\)\( \beta_{13} + \)\(18\!\cdots\!48\)\( \beta_{14} + \)\(33\!\cdots\!81\)\( \beta_{15} + \)\(27\!\cdots\!68\)\( \beta_{16} + \)\(83\!\cdots\!20\)\( \beta_{17} + \)\(40\!\cdots\!01\)\( \beta_{18} + 57775891714146952704 \beta_{19}) q^{62} +(\)\(67\!\cdots\!16\)\( - \)\(87\!\cdots\!23\)\( \beta_{1} + \)\(55\!\cdots\!39\)\( \beta_{2} - \)\(86\!\cdots\!64\)\( \beta_{3} + \)\(18\!\cdots\!31\)\( \beta_{4} - \)\(94\!\cdots\!72\)\( \beta_{5} + \)\(47\!\cdots\!13\)\( \beta_{6} + \)\(81\!\cdots\!04\)\( \beta_{7} + \)\(10\!\cdots\!94\)\( \beta_{8} + \)\(35\!\cdots\!62\)\( \beta_{9} - \)\(47\!\cdots\!92\)\( \beta_{10} + \)\(16\!\cdots\!80\)\( \beta_{11} - \)\(97\!\cdots\!70\)\( \beta_{12} + \)\(71\!\cdots\!82\)\( \beta_{13} - \)\(12\!\cdots\!96\)\( \beta_{14} - \)\(19\!\cdots\!32\)\( \beta_{15} + \)\(15\!\cdots\!54\)\( \beta_{16} - \)\(34\!\cdots\!46\)\( \beta_{17} - \)\(11\!\cdots\!66\)\( \beta_{18} + \)\(11\!\cdots\!66\)\( \beta_{19}) q^{63} +(-\)\(21\!\cdots\!36\)\( + \)\(15\!\cdots\!76\)\( \beta_{1} - \)\(16\!\cdots\!44\)\( \beta_{2} + \)\(41\!\cdots\!16\)\( \beta_{3} + \)\(11\!\cdots\!12\)\( \beta_{4} + \)\(49\!\cdots\!96\)\( \beta_{5} - \)\(67\!\cdots\!00\)\( \beta_{6} + \)\(36\!\cdots\!76\)\( \beta_{7} - \)\(10\!\cdots\!64\)\( \beta_{8} - \)\(42\!\cdots\!16\)\( \beta_{9} + \)\(18\!\cdots\!72\)\( \beta_{10} + \)\(38\!\cdots\!28\)\( \beta_{11} - \)\(53\!\cdots\!16\)\( \beta_{12} + \)\(10\!\cdots\!00\)\( \beta_{13} - \)\(50\!\cdots\!60\)\( \beta_{15} + \)\(13\!\cdots\!68\)\( \beta_{16}) q^{64} +(-\)\(13\!\cdots\!74\)\( - \)\(17\!\cdots\!86\)\( \beta_{1} + \)\(65\!\cdots\!71\)\( \beta_{2} + \)\(36\!\cdots\!28\)\( \beta_{3} + \)\(18\!\cdots\!31\)\( \beta_{4} - \)\(40\!\cdots\!97\)\( \beta_{5} + \)\(10\!\cdots\!60\)\( \beta_{6} - \)\(58\!\cdots\!96\)\( \beta_{7} + \)\(66\!\cdots\!79\)\( \beta_{8} + \)\(93\!\cdots\!64\)\( \beta_{9} + \)\(29\!\cdots\!45\)\( \beta_{10} + \)\(31\!\cdots\!66\)\( \beta_{11} - \)\(53\!\cdots\!67\)\( \beta_{12} - \)\(29\!\cdots\!64\)\( \beta_{13} - \)\(99\!\cdots\!72\)\( \beta_{14} + \)\(10\!\cdots\!61\)\( \beta_{15} + \)\(19\!\cdots\!28\)\( \beta_{16} - \)\(97\!\cdots\!26\)\( \beta_{17} + 42293463504303299158 \beta_{18} - 86790530492642506842 \beta_{19}) q^{65} +(\)\(41\!\cdots\!39\)\( + \)\(29\!\cdots\!55\)\( \beta_{1} + \)\(59\!\cdots\!74\)\( \beta_{2} - \)\(11\!\cdots\!39\)\( \beta_{3} - \)\(55\!\cdots\!27\)\( \beta_{4} - \)\(21\!\cdots\!27\)\( \beta_{5} + \)\(22\!\cdots\!69\)\( \beta_{6} - \)\(13\!\cdots\!91\)\( \beta_{7} - \)\(64\!\cdots\!82\)\( \beta_{8} + \)\(41\!\cdots\!09\)\( \beta_{9} + \)\(19\!\cdots\!34\)\( \beta_{10} + \)\(55\!\cdots\!14\)\( \beta_{11} + \)\(24\!\cdots\!30\)\( \beta_{12} - \)\(14\!\cdots\!25\)\( \beta_{13} + \)\(11\!\cdots\!84\)\( \beta_{14} + \)\(26\!\cdots\!18\)\( \beta_{15} + \)\(40\!\cdots\!14\)\( \beta_{16} - \)\(71\!\cdots\!84\)\( \beta_{17} + \)\(57\!\cdots\!53\)\( \beta_{18} - \)\(75\!\cdots\!80\)\( \beta_{19}) q^{66} +(-\)\(31\!\cdots\!50\)\( + \)\(19\!\cdots\!35\)\( \beta_{1} - \)\(20\!\cdots\!07\)\( \beta_{2} + \)\(10\!\cdots\!99\)\( \beta_{3} + \)\(60\!\cdots\!66\)\( \beta_{4} + \)\(48\!\cdots\!64\)\( \beta_{5} + \)\(16\!\cdots\!96\)\( \beta_{6} + \)\(21\!\cdots\!31\)\( \beta_{7} - \)\(13\!\cdots\!94\)\( \beta_{8} + \)\(14\!\cdots\!88\)\( \beta_{9} - \)\(61\!\cdots\!66\)\( \beta_{10} + \)\(50\!\cdots\!00\)\( \beta_{11} - \)\(13\!\cdots\!02\)\( \beta_{12} - \)\(10\!\cdots\!00\)\( \beta_{13} + \)\(26\!\cdots\!34\)\( \beta_{15} - \)\(26\!\cdots\!32\)\( \beta_{16}) q^{67} +(\)\(27\!\cdots\!08\)\( - \)\(63\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!04\)\( \beta_{2} - \)\(75\!\cdots\!00\)\( \beta_{3} + \)\(17\!\cdots\!72\)\( \beta_{4} - \)\(17\!\cdots\!68\)\( \beta_{5} + \)\(40\!\cdots\!36\)\( \beta_{6} + \)\(45\!\cdots\!44\)\( \beta_{7} + \)\(13\!\cdots\!88\)\( \beta_{8} - \)\(29\!\cdots\!68\)\( \beta_{9} + \)\(12\!\cdots\!28\)\( \beta_{10} + \)\(11\!\cdots\!64\)\( \beta_{11} + \)\(19\!\cdots\!72\)\( \beta_{12} + \)\(12\!\cdots\!08\)\( \beta_{13} + \)\(93\!\cdots\!92\)\( \beta_{14} - \)\(30\!\cdots\!36\)\( \beta_{15} - \)\(39\!\cdots\!08\)\( \beta_{16} + \)\(37\!\cdots\!20\)\( \beta_{17} - \)\(11\!\cdots\!76\)\( \beta_{18} - \)\(96\!\cdots\!64\)\( \beta_{19}) q^{68} +(\)\(13\!\cdots\!44\)\( - \)\(26\!\cdots\!74\)\( \beta_{1} + \)\(88\!\cdots\!19\)\( \beta_{2} - \)\(21\!\cdots\!56\)\( \beta_{3} - \)\(10\!\cdots\!85\)\( \beta_{4} - \)\(42\!\cdots\!45\)\( \beta_{5} - \)\(25\!\cdots\!44\)\( \beta_{6} + \)\(78\!\cdots\!70\)\( \beta_{7} - \)\(31\!\cdots\!55\)\( \beta_{8} - \)\(27\!\cdots\!24\)\( \beta_{9} + \)\(11\!\cdots\!67\)\( \beta_{10} + \)\(38\!\cdots\!32\)\( \beta_{11} - \)\(39\!\cdots\!69\)\( \beta_{12} + \)\(10\!\cdots\!96\)\( \beta_{13} - \)\(14\!\cdots\!96\)\( \beta_{14} - \)\(12\!\cdots\!87\)\( \beta_{15} - \)\(31\!\cdots\!70\)\( \beta_{16} + \)\(44\!\cdots\!18\)\( \beta_{17} - \)\(19\!\cdots\!06\)\( \beta_{18} + \)\(35\!\cdots\!70\)\( \beta_{19}) q^{69} +(-\)\(26\!\cdots\!90\)\( - \)\(18\!\cdots\!60\)\( \beta_{1} + \)\(19\!\cdots\!06\)\( \beta_{2} + \)\(73\!\cdots\!78\)\( \beta_{3} - \)\(31\!\cdots\!72\)\( \beta_{4} - \)\(34\!\cdots\!80\)\( \beta_{5} + \)\(26\!\cdots\!28\)\( \beta_{6} + \)\(93\!\cdots\!00\)\( \beta_{7} + \)\(44\!\cdots\!48\)\( \beta_{8} + \)\(80\!\cdots\!74\)\( \beta_{9} - \)\(34\!\cdots\!00\)\( \beta_{10} - \)\(26\!\cdots\!62\)\( \beta_{11} + \)\(22\!\cdots\!26\)\( \beta_{12} + \)\(35\!\cdots\!48\)\( \beta_{13} - \)\(11\!\cdots\!18\)\( \beta_{15} - \)\(54\!\cdots\!96\)\( \beta_{16}) q^{70} +(-\)\(11\!\cdots\!88\)\( + \)\(55\!\cdots\!84\)\( \beta_{1} + \)\(56\!\cdots\!43\)\( \beta_{2} + \)\(31\!\cdots\!42\)\( \beta_{3} + \)\(35\!\cdots\!17\)\( \beta_{4} + \)\(10\!\cdots\!23\)\( \beta_{5} - \)\(26\!\cdots\!04\)\( \beta_{6} + \)\(66\!\cdots\!50\)\( \beta_{7} - \)\(16\!\cdots\!11\)\( \beta_{8} - \)\(84\!\cdots\!02\)\( \beta_{9} - \)\(75\!\cdots\!59\)\( \beta_{10} - \)\(77\!\cdots\!22\)\( \beta_{11} + \)\(87\!\cdots\!63\)\( \beta_{12} - \)\(22\!\cdots\!80\)\( \beta_{13} + \)\(11\!\cdots\!06\)\( \beta_{14} - \)\(11\!\cdots\!23\)\( \beta_{15} - \)\(42\!\cdots\!98\)\( \beta_{16} - \)\(19\!\cdots\!66\)\( \beta_{17} + \)\(87\!\cdots\!02\)\( \beta_{18} + \)\(10\!\cdots\!90\)\( \beta_{19}) q^{71} +(\)\(67\!\cdots\!83\)\( - \)\(35\!\cdots\!32\)\( \beta_{1} + \)\(46\!\cdots\!42\)\( \beta_{2} - \)\(21\!\cdots\!41\)\( \beta_{3} - \)\(33\!\cdots\!91\)\( \beta_{4} + \)\(10\!\cdots\!87\)\( \beta_{5} - \)\(17\!\cdots\!10\)\( \beta_{6} + \)\(38\!\cdots\!25\)\( \beta_{7} + \)\(89\!\cdots\!97\)\( \beta_{8} + \)\(44\!\cdots\!74\)\( \beta_{9} - \)\(80\!\cdots\!88\)\( \beta_{10} - \)\(16\!\cdots\!40\)\( \beta_{11} + \)\(24\!\cdots\!48\)\( \beta_{12} - \)\(40\!\cdots\!10\)\( \beta_{13} - \)\(17\!\cdots\!76\)\( \beta_{14} + \)\(14\!\cdots\!86\)\( \beta_{15} - \)\(38\!\cdots\!42\)\( \beta_{16} - \)\(10\!\cdots\!28\)\( \beta_{17} + \)\(40\!\cdots\!96\)\( \beta_{18} - \)\(12\!\cdots\!76\)\( \beta_{19}) q^{72} +(-\)\(11\!\cdots\!82\)\( + \)\(42\!\cdots\!12\)\( \beta_{1} - \)\(45\!\cdots\!36\)\( \beta_{2} + \)\(13\!\cdots\!72\)\( \beta_{3} + \)\(22\!\cdots\!02\)\( \beta_{4} - \)\(14\!\cdots\!40\)\( \beta_{5} + \)\(12\!\cdots\!88\)\( \beta_{6} - \)\(23\!\cdots\!16\)\( \beta_{7} + \)\(52\!\cdots\!88\)\( \beta_{8} - \)\(24\!\cdots\!86\)\( \beta_{9} + \)\(89\!\cdots\!68\)\( \beta_{10} - \)\(14\!\cdots\!52\)\( \beta_{11} - \)\(89\!\cdots\!66\)\( \beta_{12} + \)\(10\!\cdots\!98\)\( \beta_{13} - \)\(79\!\cdots\!58\)\( \beta_{15} + \)\(41\!\cdots\!18\)\( \beta_{16}) q^{73} +(-\)\(18\!\cdots\!66\)\( + \)\(18\!\cdots\!50\)\( \beta_{1} + \)\(94\!\cdots\!40\)\( \beta_{2} + \)\(52\!\cdots\!44\)\( \beta_{3} - \)\(98\!\cdots\!90\)\( \beta_{4} + \)\(24\!\cdots\!14\)\( \beta_{5} - \)\(64\!\cdots\!46\)\( \beta_{6} - \)\(19\!\cdots\!62\)\( \beta_{7} - \)\(34\!\cdots\!20\)\( \beta_{8} + \)\(48\!\cdots\!58\)\( \beta_{9} - \)\(20\!\cdots\!44\)\( \beta_{10} - \)\(19\!\cdots\!88\)\( \beta_{11} - \)\(24\!\cdots\!26\)\( \beta_{12} - \)\(11\!\cdots\!70\)\( \beta_{13} - \)\(40\!\cdots\!24\)\( \beta_{14} + \)\(54\!\cdots\!26\)\( \beta_{15} + \)\(11\!\cdots\!16\)\( \beta_{16} - \)\(42\!\cdots\!68\)\( \beta_{17} - \)\(40\!\cdots\!34\)\( \beta_{18} - \)\(59\!\cdots\!20\)\( \beta_{19}) q^{74} +(\)\(61\!\cdots\!55\)\( + \)\(16\!\cdots\!70\)\( \beta_{1} + \)\(15\!\cdots\!78\)\( \beta_{2} - \)\(93\!\cdots\!86\)\( \beta_{3} + \)\(15\!\cdots\!39\)\( \beta_{4} + \)\(34\!\cdots\!10\)\( \beta_{5} + \)\(23\!\cdots\!14\)\( \beta_{6} + \)\(67\!\cdots\!25\)\( \beta_{7} + \)\(75\!\cdots\!24\)\( \beta_{8} + \)\(15\!\cdots\!62\)\( \beta_{9} + \)\(29\!\cdots\!00\)\( \beta_{10} - \)\(10\!\cdots\!31\)\( \beta_{11} - \)\(21\!\cdots\!87\)\( \beta_{12} + \)\(67\!\cdots\!74\)\( \beta_{13} + \)\(94\!\cdots\!25\)\( \beta_{14} + \)\(90\!\cdots\!41\)\( \beta_{15} + \)\(78\!\cdots\!52\)\( \beta_{16} - \)\(21\!\cdots\!00\)\( \beta_{17} - \)\(43\!\cdots\!00\)\( \beta_{18} + \)\(29\!\cdots\!00\)\( \beta_{19}) q^{75} +(-\)\(44\!\cdots\!21\)\( - \)\(78\!\cdots\!34\)\( \beta_{1} + \)\(84\!\cdots\!33\)\( \beta_{2} + \)\(14\!\cdots\!76\)\( \beta_{3} - \)\(88\!\cdots\!36\)\( \beta_{4} - \)\(64\!\cdots\!19\)\( \beta_{5} - \)\(12\!\cdots\!67\)\( \beta_{6} + \)\(29\!\cdots\!15\)\( \beta_{7} + \)\(12\!\cdots\!19\)\( \beta_{8} - \)\(11\!\cdots\!53\)\( \beta_{9} - \)\(28\!\cdots\!27\)\( \beta_{10} - \)\(32\!\cdots\!32\)\( \beta_{11} + \)\(10\!\cdots\!01\)\( \beta_{12} - \)\(10\!\cdots\!19\)\( \beta_{13} + \)\(17\!\cdots\!24\)\( \beta_{15} - \)\(59\!\cdots\!47\)\( \beta_{16}) q^{76} +(\)\(19\!\cdots\!54\)\( - \)\(39\!\cdots\!62\)\( \beta_{1} - \)\(99\!\cdots\!32\)\( \beta_{2} - \)\(55\!\cdots\!14\)\( \beta_{3} - \)\(65\!\cdots\!68\)\( \beta_{4} + \)\(24\!\cdots\!30\)\( \beta_{5} - \)\(96\!\cdots\!08\)\( \beta_{6} + \)\(42\!\cdots\!32\)\( \beta_{7} + \)\(10\!\cdots\!26\)\( \beta_{8} + \)\(77\!\cdots\!44\)\( \beta_{9} - \)\(29\!\cdots\!50\)\( \beta_{10} - \)\(28\!\cdots\!24\)\( \beta_{11} + \)\(36\!\cdots\!42\)\( \beta_{12} + \)\(77\!\cdots\!52\)\( \beta_{13} - \)\(93\!\cdots\!84\)\( \beta_{14} + \)\(11\!\cdots\!42\)\( \beta_{15} + \)\(67\!\cdots\!32\)\( \beta_{16} + \)\(14\!\cdots\!64\)\( \beta_{17} + \)\(12\!\cdots\!28\)\( \beta_{18} + \)\(22\!\cdots\!72\)\( \beta_{19}) q^{77} +(\)\(21\!\cdots\!34\)\( + \)\(54\!\cdots\!22\)\( \beta_{1} + \)\(33\!\cdots\!04\)\( \beta_{2} + \)\(16\!\cdots\!44\)\( \beta_{3} + \)\(10\!\cdots\!10\)\( \beta_{4} - \)\(73\!\cdots\!38\)\( \beta_{5} + \)\(64\!\cdots\!26\)\( \beta_{6} + \)\(71\!\cdots\!46\)\( \beta_{7} - \)\(11\!\cdots\!94\)\( \beta_{8} - \)\(73\!\cdots\!38\)\( \beta_{9} + \)\(65\!\cdots\!22\)\( \beta_{10} + \)\(51\!\cdots\!36\)\( \beta_{11} + \)\(11\!\cdots\!24\)\( \beta_{12} + \)\(28\!\cdots\!10\)\( \beta_{13} - \)\(88\!\cdots\!96\)\( \beta_{14} + \)\(37\!\cdots\!36\)\( \beta_{15} - \)\(22\!\cdots\!40\)\( \beta_{16} + \)\(24\!\cdots\!96\)\( \beta_{17} + \)\(18\!\cdots\!82\)\( \beta_{18} - \)\(22\!\cdots\!92\)\( \beta_{19}) q^{78} +(-\)\(90\!\cdots\!66\)\( - \)\(16\!\cdots\!81\)\( \beta_{1} + \)\(17\!\cdots\!62\)\( \beta_{2} - \)\(10\!\cdots\!70\)\( \beta_{3} - \)\(16\!\cdots\!28\)\( \beta_{4} + \)\(20\!\cdots\!37\)\( \beta_{5} + \)\(86\!\cdots\!21\)\( \beta_{6} + \)\(86\!\cdots\!36\)\( \beta_{7} - \)\(64\!\cdots\!03\)\( \beta_{8} + \)\(23\!\cdots\!44\)\( \beta_{9} - \)\(52\!\cdots\!85\)\( \beta_{10} + \)\(20\!\cdots\!84\)\( \beta_{11} + \)\(31\!\cdots\!53\)\( \beta_{12} + \)\(71\!\cdots\!42\)\( \beta_{13} - \)\(28\!\cdots\!27\)\( \beta_{15} - \)\(86\!\cdots\!52\)\( \beta_{16}) q^{79} +(\)\(12\!\cdots\!76\)\( - \)\(13\!\cdots\!16\)\( \beta_{1} - \)\(60\!\cdots\!44\)\( \beta_{2} - \)\(33\!\cdots\!12\)\( \beta_{3} - \)\(46\!\cdots\!84\)\( \beta_{4} - \)\(80\!\cdots\!52\)\( \beta_{5} + \)\(18\!\cdots\!00\)\( \beta_{6} + \)\(18\!\cdots\!64\)\( \beta_{7} + \)\(24\!\cdots\!24\)\( \beta_{8} - \)\(72\!\cdots\!96\)\( \beta_{9} + \)\(56\!\cdots\!60\)\( \beta_{10} + \)\(54\!\cdots\!96\)\( \beta_{11} + \)\(26\!\cdots\!68\)\( \beta_{12} + \)\(35\!\cdots\!36\)\( \beta_{13} + \)\(11\!\cdots\!68\)\( \beta_{14} - \)\(89\!\cdots\!04\)\( \beta_{15} - \)\(25\!\cdots\!92\)\( \beta_{16} + \)\(33\!\cdots\!64\)\( \beta_{17} - \)\(23\!\cdots\!92\)\( \beta_{18} - \)\(55\!\cdots\!72\)\( \beta_{19}) q^{80} +(-\)\(13\!\cdots\!20\)\( + \)\(36\!\cdots\!12\)\( \beta_{1} - \)\(28\!\cdots\!60\)\( \beta_{2} + \)\(22\!\cdots\!47\)\( \beta_{3} + \)\(17\!\cdots\!48\)\( \beta_{4} - \)\(18\!\cdots\!31\)\( \beta_{5} - \)\(94\!\cdots\!54\)\( \beta_{6} + \)\(71\!\cdots\!11\)\( \beta_{7} - \)\(21\!\cdots\!44\)\( \beta_{8} + \)\(18\!\cdots\!19\)\( \beta_{9} - \)\(58\!\cdots\!11\)\( \beta_{10} + \)\(46\!\cdots\!01\)\( \beta_{11} - \)\(32\!\cdots\!29\)\( \beta_{12} - \)\(23\!\cdots\!12\)\( \beta_{13} - \)\(76\!\cdots\!76\)\( \beta_{14} - \)\(24\!\cdots\!32\)\( \beta_{15} - \)\(33\!\cdots\!87\)\( \beta_{16} - \)\(81\!\cdots\!28\)\( \beta_{17} - \)\(22\!\cdots\!24\)\( \beta_{18} - \)\(18\!\cdots\!20\)\( \beta_{19}) q^{81} +(\)\(48\!\cdots\!72\)\( - \)\(51\!\cdots\!50\)\( \beta_{1} + \)\(55\!\cdots\!18\)\( \beta_{2} - \)\(10\!\cdots\!02\)\( \beta_{3} - \)\(37\!\cdots\!42\)\( \beta_{4} + \)\(78\!\cdots\!98\)\( \beta_{5} + \)\(11\!\cdots\!44\)\( \beta_{6} + \)\(22\!\cdots\!96\)\( \beta_{7} - \)\(26\!\cdots\!94\)\( \beta_{8} + \)\(10\!\cdots\!48\)\( \beta_{9} + \)\(46\!\cdots\!42\)\( \beta_{10} + \)\(62\!\cdots\!04\)\( \beta_{11} + \)\(80\!\cdots\!48\)\( \beta_{12} - \)\(25\!\cdots\!96\)\( \beta_{13} - \)\(13\!\cdots\!76\)\( \beta_{15} - \)\(13\!\cdots\!40\)\( \beta_{16}) q^{82} +(\)\(15\!\cdots\!39\)\( + \)\(39\!\cdots\!26\)\( \beta_{1} - \)\(76\!\cdots\!57\)\( \beta_{2} - \)\(42\!\cdots\!05\)\( \beta_{3} - \)\(16\!\cdots\!20\)\( \beta_{4} - \)\(42\!\cdots\!37\)\( \beta_{5} + \)\(56\!\cdots\!26\)\( \beta_{6} + \)\(28\!\cdots\!80\)\( \beta_{7} - \)\(11\!\cdots\!29\)\( \beta_{8} + \)\(11\!\cdots\!02\)\( \beta_{9} + \)\(99\!\cdots\!61\)\( \beta_{10} + \)\(11\!\cdots\!55\)\( \beta_{11} - \)\(16\!\cdots\!02\)\( \beta_{12} - \)\(19\!\cdots\!40\)\( \beta_{13} - \)\(33\!\cdots\!09\)\( \beta_{14} - \)\(90\!\cdots\!58\)\( \beta_{15} - \)\(84\!\cdots\!12\)\( \beta_{16} - \)\(44\!\cdots\!92\)\( \beta_{17} + \)\(13\!\cdots\!24\)\( \beta_{18} + \)\(23\!\cdots\!16\)\( \beta_{19}) q^{83} +(-\)\(14\!\cdots\!25\)\( - \)\(68\!\cdots\!28\)\( \beta_{1} - \)\(22\!\cdots\!57\)\( \beta_{2} + \)\(26\!\cdots\!68\)\( \beta_{3} - \)\(51\!\cdots\!70\)\( \beta_{4} - \)\(25\!\cdots\!07\)\( \beta_{5} - \)\(16\!\cdots\!27\)\( \beta_{6} + \)\(14\!\cdots\!41\)\( \beta_{7} + \)\(18\!\cdots\!03\)\( \beta_{8} - \)\(10\!\cdots\!27\)\( \beta_{9} - \)\(58\!\cdots\!47\)\( \beta_{10} + \)\(65\!\cdots\!76\)\( \beta_{11} + \)\(49\!\cdots\!47\)\( \beta_{12} + \)\(80\!\cdots\!35\)\( \beta_{13} + \)\(19\!\cdots\!42\)\( \beta_{14} + \)\(24\!\cdots\!46\)\( \beta_{15} + \)\(12\!\cdots\!97\)\( \beta_{16} + \)\(96\!\cdots\!22\)\( \beta_{17} + \)\(14\!\cdots\!96\)\( \beta_{18} + \)\(10\!\cdots\!40\)\( \beta_{19}) q^{84} +(\)\(78\!\cdots\!00\)\( - \)\(39\!\cdots\!84\)\( \beta_{1} + \)\(42\!\cdots\!04\)\( \beta_{2} - \)\(24\!\cdots\!24\)\( \beta_{3} - \)\(39\!\cdots\!80\)\( \beta_{4} - \)\(30\!\cdots\!00\)\( \beta_{5} + \)\(83\!\cdots\!84\)\( \beta_{6} + \)\(15\!\cdots\!84\)\( \beta_{7} + \)\(72\!\cdots\!84\)\( \beta_{8} - \)\(37\!\cdots\!48\)\( \beta_{9} + \)\(26\!\cdots\!44\)\( \beta_{10} + \)\(26\!\cdots\!24\)\( \beta_{11} - \)\(13\!\cdots\!52\)\( \beta_{12} + \)\(31\!\cdots\!04\)\( \beta_{13} + \)\(23\!\cdots\!36\)\( \beta_{15} + \)\(43\!\cdots\!92\)\( \beta_{16}) q^{85} +(\)\(15\!\cdots\!75\)\( + \)\(13\!\cdots\!28\)\( \beta_{1} - \)\(79\!\cdots\!98\)\( \beta_{2} - \)\(43\!\cdots\!57\)\( \beta_{3} + \)\(24\!\cdots\!91\)\( \beta_{4} - \)\(27\!\cdots\!75\)\( \beta_{5} - \)\(32\!\cdots\!58\)\( \beta_{6} + \)\(33\!\cdots\!22\)\( \beta_{7} - \)\(84\!\cdots\!69\)\( \beta_{8} + \)\(48\!\cdots\!15\)\( \beta_{9} - \)\(89\!\cdots\!21\)\( \beta_{10} - \)\(85\!\cdots\!21\)\( \beta_{11} + \)\(54\!\cdots\!70\)\( \beta_{12} + \)\(85\!\cdots\!50\)\( \beta_{13} + \)\(31\!\cdots\!36\)\( \beta_{14} + \)\(74\!\cdots\!90\)\( \beta_{15} + \)\(57\!\cdots\!60\)\( \beta_{16} + \)\(17\!\cdots\!80\)\( \beta_{17} + \)\(13\!\cdots\!30\)\( \beta_{18} + \)\(50\!\cdots\!20\)\( \beta_{19}) q^{86} +(-\)\(10\!\cdots\!59\)\( + \)\(35\!\cdots\!92\)\( \beta_{1} - \)\(59\!\cdots\!13\)\( \beta_{2} + \)\(17\!\cdots\!21\)\( \beta_{3} - \)\(28\!\cdots\!10\)\( \beta_{4} - \)\(50\!\cdots\!39\)\( \beta_{5} + \)\(18\!\cdots\!92\)\( \beta_{6} + \)\(32\!\cdots\!31\)\( \beta_{7} + \)\(58\!\cdots\!46\)\( \beta_{8} + \)\(49\!\cdots\!55\)\( \beta_{9} + \)\(21\!\cdots\!93\)\( \beta_{10} - \)\(31\!\cdots\!05\)\( \beta_{11} - \)\(90\!\cdots\!05\)\( \beta_{12} - \)\(16\!\cdots\!17\)\( \beta_{13} + \)\(88\!\cdots\!69\)\( \beta_{14} + \)\(16\!\cdots\!86\)\( \beta_{15} - \)\(17\!\cdots\!99\)\( \beta_{16} + \)\(58\!\cdots\!95\)\( \beta_{17} - \)\(49\!\cdots\!07\)\( \beta_{18} - \)\(27\!\cdots\!03\)\( \beta_{19}) q^{87} +(\)\(21\!\cdots\!30\)\( - \)\(26\!\cdots\!00\)\( \beta_{1} + \)\(28\!\cdots\!60\)\( \beta_{2} - \)\(61\!\cdots\!48\)\( \beta_{3} - \)\(27\!\cdots\!90\)\( \beta_{4} - \)\(18\!\cdots\!20\)\( \beta_{5} - \)\(16\!\cdots\!02\)\( \beta_{6} + \)\(93\!\cdots\!06\)\( \beta_{7} + \)\(36\!\cdots\!32\)\( \beta_{8} - \)\(84\!\cdots\!54\)\( \beta_{9} - \)\(56\!\cdots\!64\)\( \beta_{10} - \)\(10\!\cdots\!96\)\( \beta_{11} + \)\(72\!\cdots\!16\)\( \beta_{12} + \)\(14\!\cdots\!54\)\( \beta_{13} + \)\(37\!\cdots\!38\)\( \beta_{15} - \)\(23\!\cdots\!82\)\( \beta_{16}) q^{88} +(\)\(26\!\cdots\!13\)\( + \)\(53\!\cdots\!20\)\( \beta_{1} - \)\(13\!\cdots\!08\)\( \beta_{2} - \)\(72\!\cdots\!51\)\( \beta_{3} + \)\(25\!\cdots\!04\)\( \beta_{4} + \)\(13\!\cdots\!83\)\( \beta_{5} - \)\(41\!\cdots\!22\)\( \beta_{6} + \)\(67\!\cdots\!57\)\( \beta_{7} + \)\(17\!\cdots\!00\)\( \beta_{8} + \)\(61\!\cdots\!89\)\( \beta_{9} - \)\(12\!\cdots\!73\)\( \beta_{10} - \)\(12\!\cdots\!65\)\( \beta_{11} - \)\(13\!\cdots\!79\)\( \beta_{12} - \)\(14\!\cdots\!16\)\( \beta_{13} - \)\(19\!\cdots\!28\)\( \beta_{14} + \)\(30\!\cdots\!64\)\( \beta_{15} - \)\(53\!\cdots\!85\)\( \beta_{16} - \)\(34\!\cdots\!24\)\( \beta_{17} - \)\(38\!\cdots\!72\)\( \beta_{18} - \)\(26\!\cdots\!80\)\( \beta_{19}) q^{89} +(-\)\(34\!\cdots\!70\)\( + \)\(53\!\cdots\!11\)\( \beta_{1} + \)\(67\!\cdots\!53\)\( \beta_{2} + \)\(77\!\cdots\!53\)\( \beta_{3} - \)\(92\!\cdots\!93\)\( \beta_{4} + \)\(44\!\cdots\!65\)\( \beta_{5} + \)\(98\!\cdots\!56\)\( \beta_{6} + \)\(14\!\cdots\!24\)\( \beta_{7} - \)\(12\!\cdots\!09\)\( \beta_{8} - \)\(80\!\cdots\!42\)\( \beta_{9} - \)\(13\!\cdots\!71\)\( \beta_{10} - \)\(21\!\cdots\!44\)\( \beta_{11} + \)\(12\!\cdots\!32\)\( \beta_{12} + \)\(24\!\cdots\!26\)\( \beta_{13} - \)\(59\!\cdots\!40\)\( \beta_{14} - \)\(32\!\cdots\!76\)\( \beta_{15} - \)\(18\!\cdots\!52\)\( \beta_{16} + \)\(90\!\cdots\!80\)\( \beta_{17} + \)\(71\!\cdots\!10\)\( \beta_{18} + \)\(32\!\cdots\!60\)\( \beta_{19}) q^{90} +(-\)\(83\!\cdots\!34\)\( - \)\(17\!\cdots\!98\)\( \beta_{1} + \)\(19\!\cdots\!01\)\( \beta_{2} - \)\(17\!\cdots\!90\)\( \beta_{3} - \)\(16\!\cdots\!77\)\( \beta_{4} - \)\(88\!\cdots\!33\)\( \beta_{5} + \)\(27\!\cdots\!88\)\( \beta_{6} + \)\(57\!\cdots\!74\)\( \beta_{7} + \)\(13\!\cdots\!27\)\( \beta_{8} + \)\(11\!\cdots\!96\)\( \beta_{9} + \)\(33\!\cdots\!97\)\( \beta_{10} - \)\(71\!\cdots\!80\)\( \beta_{11} - \)\(15\!\cdots\!77\)\( \beta_{12} - \)\(78\!\cdots\!54\)\( \beta_{13} - \)\(12\!\cdots\!41\)\( \beta_{15} + \)\(91\!\cdots\!12\)\( \beta_{16}) q^{91} +(\)\(15\!\cdots\!38\)\( - \)\(18\!\cdots\!72\)\( \beta_{1} - \)\(79\!\cdots\!74\)\( \beta_{2} - \)\(43\!\cdots\!16\)\( \beta_{3} + \)\(59\!\cdots\!80\)\( \beta_{4} - \)\(67\!\cdots\!30\)\( \beta_{5} - \)\(15\!\cdots\!30\)\( \beta_{6} + \)\(32\!\cdots\!02\)\( \beta_{7} + \)\(39\!\cdots\!98\)\( \beta_{8} - \)\(68\!\cdots\!90\)\( \beta_{9} - \)\(26\!\cdots\!06\)\( \beta_{10} - \)\(39\!\cdots\!72\)\( \beta_{11} + \)\(25\!\cdots\!38\)\( \beta_{12} + \)\(44\!\cdots\!46\)\( \beta_{13} + \)\(74\!\cdots\!12\)\( \beta_{14} - \)\(82\!\cdots\!56\)\( \beta_{15} - \)\(22\!\cdots\!82\)\( \beta_{16} + \)\(33\!\cdots\!44\)\( \beta_{17} + \)\(21\!\cdots\!20\)\( \beta_{18} + \)\(69\!\cdots\!00\)\( \beta_{19}) q^{92} +(\)\(25\!\cdots\!30\)\( + \)\(60\!\cdots\!81\)\( \beta_{1} - \)\(25\!\cdots\!93\)\( \beta_{2} - \)\(31\!\cdots\!06\)\( \beta_{3} - \)\(15\!\cdots\!06\)\( \beta_{4} + \)\(24\!\cdots\!93\)\( \beta_{5} + \)\(28\!\cdots\!64\)\( \beta_{6} + \)\(22\!\cdots\!57\)\( \beta_{7} - \)\(14\!\cdots\!10\)\( \beta_{8} - \)\(89\!\cdots\!48\)\( \beta_{9} - \)\(88\!\cdots\!47\)\( \beta_{10} + \)\(15\!\cdots\!05\)\( \beta_{11} - \)\(21\!\cdots\!40\)\( \beta_{12} - \)\(45\!\cdots\!13\)\( \beta_{13} + \)\(11\!\cdots\!24\)\( \beta_{14} - \)\(16\!\cdots\!57\)\( \beta_{15} + \)\(36\!\cdots\!74\)\( \beta_{16} - \)\(97\!\cdots\!56\)\( \beta_{17} + \)\(22\!\cdots\!84\)\( \beta_{18} + \)\(53\!\cdots\!76\)\( \beta_{19}) q^{93} +(-\)\(12\!\cdots\!84\)\( - \)\(14\!\cdots\!40\)\( \beta_{1} + \)\(15\!\cdots\!84\)\( \beta_{2} + \)\(89\!\cdots\!16\)\( \beta_{3} - \)\(16\!\cdots\!88\)\( \beta_{4} + \)\(66\!\cdots\!04\)\( \beta_{5} - \)\(21\!\cdots\!20\)\( \beta_{6} + \)\(67\!\cdots\!80\)\( \beta_{7} - \)\(13\!\cdots\!32\)\( \beta_{8} + \)\(67\!\cdots\!56\)\( \beta_{9} + \)\(44\!\cdots\!12\)\( \beta_{10} + \)\(24\!\cdots\!40\)\( \beta_{11} - \)\(64\!\cdots\!48\)\( \beta_{12} + \)\(17\!\cdots\!00\)\( \beta_{13} - \)\(30\!\cdots\!40\)\( \beta_{15} - \)\(24\!\cdots\!16\)\( \beta_{16}) q^{94} +(\)\(33\!\cdots\!34\)\( + \)\(24\!\cdots\!50\)\( \beta_{1} - \)\(16\!\cdots\!41\)\( \beta_{2} - \)\(94\!\cdots\!46\)\( \beta_{3} - \)\(51\!\cdots\!87\)\( \beta_{4} - \)\(86\!\cdots\!07\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(62\!\cdots\!24\)\( \beta_{7} - \)\(77\!\cdots\!41\)\( \beta_{8} - \)\(12\!\cdots\!86\)\( \beta_{9} + \)\(33\!\cdots\!35\)\( \beta_{10} + \)\(33\!\cdots\!36\)\( \beta_{11} - \)\(34\!\cdots\!37\)\( \beta_{12} - \)\(34\!\cdots\!24\)\( \beta_{13} - \)\(44\!\cdots\!12\)\( \beta_{14} + \)\(15\!\cdots\!61\)\( \beta_{15} - \)\(98\!\cdots\!22\)\( \beta_{16} - \)\(84\!\cdots\!26\)\( \beta_{17} + \)\(48\!\cdots\!78\)\( \beta_{18} - \)\(61\!\cdots\!02\)\( \beta_{19}) q^{95} +(\)\(85\!\cdots\!08\)\( - \)\(26\!\cdots\!04\)\( \beta_{1} + \)\(51\!\cdots\!12\)\( \beta_{2} - \)\(20\!\cdots\!16\)\( \beta_{3} + \)\(13\!\cdots\!48\)\( \beta_{4} - \)\(24\!\cdots\!96\)\( \beta_{5} - \)\(19\!\cdots\!80\)\( \beta_{6} + \)\(14\!\cdots\!12\)\( \beta_{7} + \)\(99\!\cdots\!20\)\( \beta_{8} + \)\(69\!\cdots\!16\)\( \beta_{9} - \)\(43\!\cdots\!32\)\( \beta_{10} + \)\(15\!\cdots\!72\)\( \beta_{11} + \)\(10\!\cdots\!56\)\( \beta_{12} + \)\(71\!\cdots\!60\)\( \beta_{13} + \)\(23\!\cdots\!12\)\( \beta_{14} + \)\(74\!\cdots\!48\)\( \beta_{15} - \)\(30\!\cdots\!24\)\( \beta_{16} + \)\(36\!\cdots\!36\)\( \beta_{17} - \)\(16\!\cdots\!52\)\( \beta_{18} - \)\(28\!\cdots\!40\)\( \beta_{19}) q^{96} +(-\)\(10\!\cdots\!13\)\( - \)\(25\!\cdots\!26\)\( \beta_{1} + \)\(27\!\cdots\!86\)\( \beta_{2} + \)\(15\!\cdots\!31\)\( \beta_{3} - \)\(29\!\cdots\!74\)\( \beta_{4} + \)\(15\!\cdots\!11\)\( \beta_{5} + \)\(84\!\cdots\!10\)\( \beta_{6} + \)\(13\!\cdots\!27\)\( \beta_{7} - \)\(44\!\cdots\!02\)\( \beta_{8} - \)\(20\!\cdots\!11\)\( \beta_{9} - \)\(10\!\cdots\!83\)\( \beta_{10} + \)\(19\!\cdots\!05\)\( \beta_{11} + \)\(10\!\cdots\!79\)\( \beta_{12} - \)\(22\!\cdots\!20\)\( \beta_{13} + \)\(60\!\cdots\!12\)\( \beta_{15} + \)\(29\!\cdots\!49\)\( \beta_{16}) q^{97} +(\)\(98\!\cdots\!50\)\( + \)\(12\!\cdots\!59\)\( \beta_{1} - \)\(49\!\cdots\!20\)\( \beta_{2} - \)\(27\!\cdots\!84\)\( \beta_{3} - \)\(12\!\cdots\!58\)\( \beta_{4} - \)\(60\!\cdots\!78\)\( \beta_{5} + \)\(13\!\cdots\!66\)\( \beta_{6} + \)\(15\!\cdots\!82\)\( \beta_{7} - \)\(18\!\cdots\!00\)\( \beta_{8} + \)\(65\!\cdots\!62\)\( \beta_{9} + \)\(39\!\cdots\!04\)\( \beta_{10} + \)\(40\!\cdots\!96\)\( \beta_{11} + \)\(12\!\cdots\!34\)\( \beta_{12} + \)\(36\!\cdots\!42\)\( \beta_{13} + \)\(11\!\cdots\!40\)\( \beta_{14} + \)\(73\!\cdots\!10\)\( \beta_{15} + \)\(37\!\cdots\!04\)\( \beta_{16} + \)\(62\!\cdots\!76\)\( \beta_{17} + \)\(56\!\cdots\!14\)\( \beta_{18} - \)\(31\!\cdots\!44\)\( \beta_{19}) q^{98} +(\)\(65\!\cdots\!98\)\( - \)\(39\!\cdots\!77\)\( \beta_{1} - \)\(35\!\cdots\!32\)\( \beta_{2} - \)\(13\!\cdots\!71\)\( \beta_{3} + \)\(76\!\cdots\!13\)\( \beta_{4} + \)\(38\!\cdots\!31\)\( \beta_{5} + \)\(20\!\cdots\!38\)\( \beta_{6} + \)\(13\!\cdots\!57\)\( \beta_{7} + \)\(17\!\cdots\!31\)\( \beta_{8} + \)\(23\!\cdots\!46\)\( \beta_{9} + \)\(41\!\cdots\!11\)\( \beta_{10} + \)\(19\!\cdots\!30\)\( \beta_{11} - \)\(22\!\cdots\!95\)\( \beta_{12} + \)\(15\!\cdots\!24\)\( \beta_{13} - \)\(91\!\cdots\!78\)\( \beta_{14} - \)\(98\!\cdots\!41\)\( \beta_{15} + \)\(11\!\cdots\!38\)\( \beta_{16} - \)\(55\!\cdots\!74\)\( \beta_{17} + \)\(47\!\cdots\!58\)\( \beta_{18} + \)\(30\!\cdots\!50\)\( \beta_{19}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4870975596954516q^{3} - \)\(87\!\cdots\!80\)\(q^{4} - \)\(36\!\cdots\!80\)\(q^{6} + \)\(12\!\cdots\!88\)\(q^{7} + \)\(15\!\cdots\!00\)\(q^{9} + O(q^{10}) \) \( 20q + 4870975596954516q^{3} - \)\(87\!\cdots\!80\)\(q^{4} - \)\(36\!\cdots\!80\)\(q^{6} + \)\(12\!\cdots\!88\)\(q^{7} + \)\(15\!\cdots\!00\)\(q^{9} - \)\(78\!\cdots\!00\)\(q^{10} + \)\(72\!\cdots\!64\)\(q^{12} + \)\(18\!\cdots\!12\)\(q^{13} + \)\(29\!\cdots\!00\)\(q^{15} + \)\(35\!\cdots\!60\)\(q^{16} - \)\(13\!\cdots\!20\)\(q^{18} - \)\(21\!\cdots\!00\)\(q^{19} + \)\(84\!\cdots\!20\)\(q^{21} - \)\(19\!\cdots\!20\)\(q^{22} - \)\(85\!\cdots\!60\)\(q^{24} - \)\(30\!\cdots\!00\)\(q^{25} + \)\(55\!\cdots\!64\)\(q^{27} + \)\(14\!\cdots\!92\)\(q^{28} - \)\(13\!\cdots\!00\)\(q^{30} + \)\(61\!\cdots\!40\)\(q^{31} + \)\(76\!\cdots\!60\)\(q^{33} - \)\(60\!\cdots\!60\)\(q^{34} - \)\(31\!\cdots\!60\)\(q^{36} + \)\(27\!\cdots\!28\)\(q^{37} + \)\(55\!\cdots\!80\)\(q^{39} - \)\(16\!\cdots\!00\)\(q^{40} - \)\(36\!\cdots\!60\)\(q^{42} + \)\(29\!\cdots\!32\)\(q^{43} + \)\(28\!\cdots\!00\)\(q^{45} - \)\(93\!\cdots\!60\)\(q^{46} + \)\(44\!\cdots\!96\)\(q^{48} + \)\(35\!\cdots\!80\)\(q^{49} + \)\(24\!\cdots\!20\)\(q^{51} - \)\(70\!\cdots\!12\)\(q^{52} - \)\(38\!\cdots\!80\)\(q^{54} + \)\(13\!\cdots\!00\)\(q^{55} + \)\(14\!\cdots\!68\)\(q^{57} - \)\(10\!\cdots\!20\)\(q^{58} + \)\(18\!\cdots\!00\)\(q^{60} + \)\(36\!\cdots\!40\)\(q^{61} + \)\(13\!\cdots\!72\)\(q^{63} - \)\(42\!\cdots\!40\)\(q^{64} + \)\(83\!\cdots\!00\)\(q^{66} - \)\(62\!\cdots\!32\)\(q^{67} + \)\(26\!\cdots\!20\)\(q^{69} - \)\(53\!\cdots\!00\)\(q^{70} + \)\(13\!\cdots\!20\)\(q^{72} - \)\(23\!\cdots\!28\)\(q^{73} + \)\(12\!\cdots\!00\)\(q^{75} - \)\(88\!\cdots\!20\)\(q^{76} + \)\(42\!\cdots\!00\)\(q^{78} - \)\(18\!\cdots\!00\)\(q^{79} - \)\(26\!\cdots\!80\)\(q^{81} + \)\(97\!\cdots\!20\)\(q^{82} - \)\(29\!\cdots\!20\)\(q^{84} + \)\(15\!\cdots\!00\)\(q^{85} - \)\(20\!\cdots\!00\)\(q^{87} + \)\(43\!\cdots\!80\)\(q^{88} - \)\(68\!\cdots\!00\)\(q^{90} - \)\(16\!\cdots\!60\)\(q^{91} + \)\(50\!\cdots\!32\)\(q^{93} - \)\(24\!\cdots\!60\)\(q^{94} + \)\(17\!\cdots\!60\)\(q^{96} - \)\(20\!\cdots\!52\)\(q^{97} + \)\(13\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 906717684887249855 x^{18} + \)\(34\!\cdots\!60\)\( x^{16} + \)\(71\!\cdots\!60\)\( x^{14} + \)\(89\!\cdots\!80\)\( x^{12} + \)\(68\!\cdots\!76\)\( x^{10} + \)\(31\!\cdots\!00\)\( x^{8} + \)\(84\!\cdots\!00\)\( x^{6} + \)\(11\!\cdots\!00\)\( x^{4} + \)\(60\!\cdots\!00\)\( x^{2} + \)\(56\!\cdots\!00\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 36 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(70\!\cdots\!75\)\( \nu^{19} + \)\(30\!\cdots\!44\)\( \nu^{18} - \)\(64\!\cdots\!25\)\( \nu^{17} + \)\(27\!\cdots\!20\)\( \nu^{16} - \)\(24\!\cdots\!00\)\( \nu^{15} + \)\(10\!\cdots\!80\)\( \nu^{14} - \)\(50\!\cdots\!00\)\( \nu^{13} + \)\(21\!\cdots\!00\)\( \nu^{12} - \)\(63\!\cdots\!00\)\( \nu^{11} + \)\(26\!\cdots\!60\)\( \nu^{10} - \)\(48\!\cdots\!00\)\( \nu^{9} + \)\(20\!\cdots\!04\)\( \nu^{8} - \)\(22\!\cdots\!00\)\( \nu^{7} + \)\(95\!\cdots\!40\)\( \nu^{6} - \)\(59\!\cdots\!00\)\( \nu^{5} + \)\(24\!\cdots\!00\)\( \nu^{4} - \)\(81\!\cdots\!00\)\( \nu^{3} + \)\(24\!\cdots\!00\)\( \nu^{2} - \)\(42\!\cdots\!00\)\( \nu + \)\(11\!\cdots\!00\)\(\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(70\!\cdots\!75\)\( \nu^{19} + \)\(30\!\cdots\!44\)\( \nu^{18} - \)\(64\!\cdots\!25\)\( \nu^{17} + \)\(27\!\cdots\!20\)\( \nu^{16} - \)\(24\!\cdots\!00\)\( \nu^{15} + \)\(10\!\cdots\!80\)\( \nu^{14} - \)\(50\!\cdots\!00\)\( \nu^{13} + \)\(21\!\cdots\!00\)\( \nu^{12} - \)\(63\!\cdots\!00\)\( \nu^{11} + \)\(26\!\cdots\!60\)\( \nu^{10} - \)\(48\!\cdots\!00\)\( \nu^{9} + \)\(20\!\cdots\!04\)\( \nu^{8} - \)\(22\!\cdots\!00\)\( \nu^{7} + \)\(95\!\cdots\!40\)\( \nu^{6} - \)\(59\!\cdots\!00\)\( \nu^{5} + \)\(24\!\cdots\!00\)\( \nu^{4} - \)\(81\!\cdots\!00\)\( \nu^{3} + \)\(31\!\cdots\!00\)\( \nu^{2} - \)\(42\!\cdots\!00\)\( \nu + \)\(59\!\cdots\!00\)\(\)\()/ \)\(49\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(44\!\cdots\!05\)\( \nu^{19} - \)\(74\!\cdots\!52\)\( \nu^{18} + \)\(40\!\cdots\!75\)\( \nu^{17} - \)\(67\!\cdots\!60\)\( \nu^{16} + \)\(15\!\cdots\!00\)\( \nu^{15} - \)\(25\!\cdots\!40\)\( \nu^{14} + \)\(31\!\cdots\!00\)\( \nu^{13} - \)\(52\!\cdots\!00\)\( \nu^{12} + \)\(39\!\cdots\!00\)\( \nu^{11} - \)\(65\!\cdots\!80\)\( \nu^{10} + \)\(30\!\cdots\!80\)\( \nu^{9} - \)\(50\!\cdots\!32\)\( \nu^{8} + \)\(14\!\cdots\!00\)\( \nu^{7} - \)\(23\!\cdots\!20\)\( \nu^{6} + \)\(38\!\cdots\!00\)\( \nu^{5} - \)\(61\!\cdots\!00\)\( \nu^{4} + \)\(52\!\cdots\!00\)\( \nu^{3} - \)\(62\!\cdots\!00\)\( \nu^{2} + \)\(28\!\cdots\!00\)\( \nu - \)\(42\!\cdots\!00\)\(\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(47\!\cdots\!59\)\( \nu^{19} + \)\(39\!\cdots\!52\)\( \nu^{18} + \)\(42\!\cdots\!45\)\( \nu^{17} + \)\(30\!\cdots\!60\)\( \nu^{16} + \)\(16\!\cdots\!80\)\( \nu^{15} + \)\(11\!\cdots\!40\)\( \nu^{14} + \)\(33\!\cdots\!00\)\( \nu^{13} + \)\(33\!\cdots\!00\)\( \nu^{12} + \)\(42\!\cdots\!60\)\( \nu^{11} + \)\(64\!\cdots\!80\)\( \nu^{10} + \)\(32\!\cdots\!44\)\( \nu^{9} + \)\(75\!\cdots\!32\)\( \nu^{8} + \)\(15\!\cdots\!40\)\( \nu^{7} + \)\(48\!\cdots\!20\)\( \nu^{6} + \)\(40\!\cdots\!00\)\( \nu^{5} + \)\(14\!\cdots\!00\)\( \nu^{4} + \)\(55\!\cdots\!00\)\( \nu^{3} + \)\(12\!\cdots\!00\)\( \nu^{2} + \)\(29\!\cdots\!00\)\( \nu - \)\(54\!\cdots\!00\)\(\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(33\!\cdots\!31\)\( \nu^{19} + \)\(49\!\cdots\!48\)\( \nu^{18} - \)\(30\!\cdots\!05\)\( \nu^{17} + \)\(45\!\cdots\!20\)\( \nu^{16} - \)\(11\!\cdots\!20\)\( \nu^{15} + \)\(17\!\cdots\!80\)\( \nu^{14} - \)\(23\!\cdots\!00\)\( \nu^{13} + \)\(37\!\cdots\!80\)\( \nu^{12} - \)\(29\!\cdots\!40\)\( \nu^{11} + \)\(46\!\cdots\!40\)\( \nu^{10} - \)\(22\!\cdots\!96\)\( \nu^{9} + \)\(34\!\cdots\!48\)\( \nu^{8} - \)\(10\!\cdots\!60\)\( \nu^{7} + \)\(14\!\cdots\!80\)\( \nu^{6} - \)\(28\!\cdots\!00\)\( \nu^{5} + \)\(33\!\cdots\!00\)\( \nu^{4} - \)\(38\!\cdots\!00\)\( \nu^{3} + \)\(26\!\cdots\!00\)\( \nu^{2} - \)\(20\!\cdots\!00\)\( \nu + \)\(80\!\cdots\!00\)\(\)\()/ \)\(68\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(93\!\cdots\!67\)\( \nu^{19} + \)\(38\!\cdots\!20\)\( \nu^{18} + \)\(84\!\cdots\!85\)\( \nu^{17} + \)\(34\!\cdots\!00\)\( \nu^{16} + \)\(32\!\cdots\!40\)\( \nu^{15} + \)\(13\!\cdots\!00\)\( \nu^{14} + \)\(67\!\cdots\!00\)\( \nu^{13} + \)\(27\!\cdots\!00\)\( \nu^{12} + \)\(83\!\cdots\!80\)\( \nu^{11} + \)\(34\!\cdots\!00\)\( \nu^{10} + \)\(63\!\cdots\!72\)\( \nu^{9} + \)\(27\!\cdots\!20\)\( \nu^{8} + \)\(29\!\cdots\!20\)\( \nu^{7} + \)\(13\!\cdots\!00\)\( \nu^{6} + \)\(79\!\cdots\!00\)\( \nu^{5} + \)\(34\!\cdots\!00\)\( \nu^{4} + \)\(10\!\cdots\!00\)\( \nu^{3} + \)\(34\!\cdots\!00\)\( \nu^{2} + \)\(56\!\cdots\!00\)\( \nu + \)\(19\!\cdots\!00\)\(\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(19\!\cdots\!11\)\( \nu^{19} - \)\(26\!\cdots\!16\)\( \nu^{18} - \)\(17\!\cdots\!05\)\( \nu^{17} - \)\(21\!\cdots\!80\)\( \nu^{16} - \)\(66\!\cdots\!20\)\( \nu^{15} - \)\(84\!\cdots\!20\)\( \nu^{14} - \)\(13\!\cdots\!00\)\( \nu^{13} - \)\(20\!\cdots\!00\)\( \nu^{12} - \)\(17\!\cdots\!40\)\( \nu^{11} - \)\(33\!\cdots\!40\)\( \nu^{10} - \)\(13\!\cdots\!76\)\( \nu^{9} - \)\(34\!\cdots\!56\)\( \nu^{8} - \)\(61\!\cdots\!60\)\( \nu^{7} - \)\(20\!\cdots\!60\)\( \nu^{6} - \)\(16\!\cdots\!00\)\( \nu^{5} - \)\(59\!\cdots\!00\)\( \nu^{4} - \)\(22\!\cdots\!00\)\( \nu^{3} - \)\(55\!\cdots\!00\)\( \nu^{2} - \)\(11\!\cdots\!00\)\( \nu + \)\(11\!\cdots\!00\)\(\)\()/ \)\(47\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(55\!\cdots\!33\)\( \nu^{19} - \)\(16\!\cdots\!60\)\( \nu^{18} - \)\(50\!\cdots\!55\)\( \nu^{17} - \)\(14\!\cdots\!80\)\( \nu^{16} - \)\(19\!\cdots\!20\)\( \nu^{15} - \)\(52\!\cdots\!20\)\( \nu^{14} - \)\(40\!\cdots\!40\)\( \nu^{13} - \)\(10\!\cdots\!80\)\( \nu^{12} - \)\(50\!\cdots\!80\)\( \nu^{11} - \)\(11\!\cdots\!20\)\( \nu^{10} - \)\(39\!\cdots\!68\)\( \nu^{9} - \)\(80\!\cdots\!40\)\( \nu^{8} - \)\(18\!\cdots\!80\)\( \nu^{7} - \)\(31\!\cdots\!00\)\( \nu^{6} - \)\(50\!\cdots\!00\)\( \nu^{5} - \)\(62\!\cdots\!00\)\( \nu^{4} - \)\(69\!\cdots\!00\)\( \nu^{3} - \)\(47\!\cdots\!00\)\( \nu^{2} - \)\(37\!\cdots\!00\)\( \nu - \)\(96\!\cdots\!00\)\(\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(37\!\cdots\!99\)\( \nu^{19} + \)\(26\!\cdots\!72\)\( \nu^{18} - \)\(33\!\cdots\!45\)\( \nu^{17} - \)\(17\!\cdots\!40\)\( \nu^{16} - \)\(12\!\cdots\!80\)\( \nu^{15} - \)\(26\!\cdots\!60\)\( \nu^{14} - \)\(26\!\cdots\!00\)\( \nu^{13} - \)\(10\!\cdots\!00\)\( \nu^{12} - \)\(33\!\cdots\!60\)\( \nu^{11} - \)\(19\!\cdots\!20\)\( \nu^{10} - \)\(25\!\cdots\!84\)\( \nu^{9} - \)\(19\!\cdots\!48\)\( \nu^{8} - \)\(11\!\cdots\!40\)\( \nu^{7} - \)\(11\!\cdots\!80\)\( \nu^{6} - \)\(31\!\cdots\!00\)\( \nu^{5} - \)\(29\!\cdots\!00\)\( \nu^{4} - \)\(42\!\cdots\!00\)\( \nu^{3} - \)\(26\!\cdots\!00\)\( \nu^{2} - \)\(22\!\cdots\!00\)\( \nu - \)\(70\!\cdots\!00\)\(\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(84\!\cdots\!43\)\( \nu^{19} - \)\(35\!\cdots\!24\)\( \nu^{18} - \)\(76\!\cdots\!25\)\( \nu^{17} - \)\(21\!\cdots\!40\)\( \nu^{16} - \)\(29\!\cdots\!00\)\( \nu^{15} - \)\(18\!\cdots\!60\)\( \nu^{14} - \)\(60\!\cdots\!60\)\( \nu^{13} + \)\(15\!\cdots\!80\)\( \nu^{12} - \)\(75\!\cdots\!60\)\( \nu^{11} + \)\(52\!\cdots\!60\)\( \nu^{10} - \)\(57\!\cdots\!48\)\( \nu^{9} + \)\(71\!\cdots\!96\)\( \nu^{8} - \)\(26\!\cdots\!80\)\( \nu^{7} + \)\(47\!\cdots\!60\)\( \nu^{6} - \)\(71\!\cdots\!00\)\( \nu^{5} + \)\(14\!\cdots\!00\)\( \nu^{4} - \)\(97\!\cdots\!00\)\( \nu^{3} + \)\(13\!\cdots\!00\)\( \nu^{2} - \)\(51\!\cdots\!00\)\( \nu - \)\(83\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(34\!\cdots\!15\)\( \nu^{19} - \)\(35\!\cdots\!12\)\( \nu^{18} + \)\(31\!\cdots\!65\)\( \nu^{17} - \)\(29\!\cdots\!60\)\( \nu^{16} + \)\(12\!\cdots\!60\)\( \nu^{15} - \)\(99\!\cdots\!40\)\( \nu^{14} + \)\(25\!\cdots\!40\)\( \nu^{13} - \)\(18\!\cdots\!00\)\( \nu^{12} + \)\(31\!\cdots\!60\)\( \nu^{11} - \)\(18\!\cdots\!80\)\( \nu^{10} + \)\(24\!\cdots\!80\)\( \nu^{9} - \)\(11\!\cdots\!92\)\( \nu^{8} + \)\(11\!\cdots\!00\)\( \nu^{7} - \)\(38\!\cdots\!20\)\( \nu^{6} + \)\(31\!\cdots\!00\)\( \nu^{5} - \)\(66\!\cdots\!00\)\( \nu^{4} + \)\(43\!\cdots\!00\)\( \nu^{3} - \)\(45\!\cdots\!00\)\( \nu^{2} + \)\(23\!\cdots\!00\)\( \nu - \)\(57\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(55\!\cdots\!03\)\( \nu^{19} + \)\(84\!\cdots\!64\)\( \nu^{18} - \)\(50\!\cdots\!65\)\( \nu^{17} + \)\(74\!\cdots\!60\)\( \nu^{16} - \)\(19\!\cdots\!60\)\( \nu^{15} + \)\(27\!\cdots\!40\)\( \nu^{14} - \)\(40\!\cdots\!00\)\( \nu^{13} + \)\(54\!\cdots\!40\)\( \nu^{12} - \)\(51\!\cdots\!20\)\( \nu^{11} + \)\(62\!\cdots\!20\)\( \nu^{10} - \)\(39\!\cdots\!48\)\( \nu^{9} + \)\(43\!\cdots\!64\)\( \nu^{8} - \)\(18\!\cdots\!80\)\( \nu^{7} + \)\(17\!\cdots\!40\)\( \nu^{6} - \)\(50\!\cdots\!00\)\( \nu^{5} + \)\(34\!\cdots\!00\)\( \nu^{4} - \)\(70\!\cdots\!00\)\( \nu^{3} + \)\(25\!\cdots\!00\)\( \nu^{2} - \)\(37\!\cdots\!00\)\( \nu - \)\(85\!\cdots\!00\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(28\!\cdots\!69\)\( \nu^{19} + \)\(26\!\cdots\!48\)\( \nu^{18} + \)\(26\!\cdots\!35\)\( \nu^{17} + \)\(23\!\cdots\!20\)\( \nu^{16} + \)\(99\!\cdots\!40\)\( \nu^{15} + \)\(83\!\cdots\!80\)\( \nu^{14} + \)\(20\!\cdots\!40\)\( \nu^{13} + \)\(16\!\cdots\!80\)\( \nu^{12} + \)\(25\!\cdots\!20\)\( \nu^{11} + \)\(18\!\cdots\!40\)\( \nu^{10} + \)\(19\!\cdots\!44\)\( \nu^{9} + \)\(12\!\cdots\!48\)\( \nu^{8} + \)\(91\!\cdots\!40\)\( \nu^{7} + \)\(48\!\cdots\!80\)\( \nu^{6} + \)\(24\!\cdots\!00\)\( \nu^{5} + \)\(96\!\cdots\!00\)\( \nu^{4} + \)\(33\!\cdots\!00\)\( \nu^{3} + \)\(72\!\cdots\!00\)\( \nu^{2} + \)\(17\!\cdots\!00\)\( \nu + \)\(15\!\cdots\!00\)\(\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(20\!\cdots\!87\)\( \nu^{19} + \)\(12\!\cdots\!16\)\( \nu^{18} - \)\(18\!\cdots\!85\)\( \nu^{17} + \)\(10\!\cdots\!40\)\( \nu^{16} - \)\(70\!\cdots\!40\)\( \nu^{15} + \)\(38\!\cdots\!60\)\( \nu^{14} - \)\(14\!\cdots\!00\)\( \nu^{13} + \)\(73\!\cdots\!60\)\( \nu^{12} - \)\(18\!\cdots\!80\)\( \nu^{11} + \)\(82\!\cdots\!80\)\( \nu^{10} - \)\(14\!\cdots\!92\)\( \nu^{9} + \)\(54\!\cdots\!16\)\( \nu^{8} - \)\(67\!\cdots\!20\)\( \nu^{7} + \)\(20\!\cdots\!60\)\( \nu^{6} - \)\(18\!\cdots\!00\)\( \nu^{5} + \)\(40\!\cdots\!00\)\( \nu^{4} - \)\(25\!\cdots\!00\)\( \nu^{3} + \)\(30\!\cdots\!00\)\( \nu^{2} - \)\(13\!\cdots\!00\)\( \nu + \)\(78\!\cdots\!00\)\(\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(42\!\cdots\!01\)\( \nu^{19} - \)\(52\!\cdots\!64\)\( \nu^{18} - \)\(39\!\cdots\!95\)\( \nu^{17} - \)\(39\!\cdots\!80\)\( \nu^{16} - \)\(15\!\cdots\!80\)\( \nu^{15} - \)\(11\!\cdots\!20\)\( \nu^{14} - \)\(31\!\cdots\!40\)\( \nu^{13} - \)\(14\!\cdots\!60\)\( \nu^{12} - \)\(39\!\cdots\!00\)\( \nu^{11} - \)\(79\!\cdots\!00\)\( \nu^{10} - \)\(30\!\cdots\!56\)\( \nu^{9} + \)\(17\!\cdots\!16\)\( \nu^{8} - \)\(14\!\cdots\!60\)\( \nu^{7} + \)\(37\!\cdots\!60\)\( \nu^{6} - \)\(39\!\cdots\!00\)\( \nu^{5} + \)\(14\!\cdots\!00\)\( \nu^{4} - \)\(54\!\cdots\!00\)\( \nu^{3} + \)\(16\!\cdots\!00\)\( \nu^{2} - \)\(29\!\cdots\!00\)\( \nu + \)\(13\!\cdots\!00\)\(\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(99\!\cdots\!49\)\( \nu^{19} - \)\(14\!\cdots\!32\)\( \nu^{18} - \)\(89\!\cdots\!75\)\( \nu^{17} - \)\(13\!\cdots\!00\)\( \nu^{16} - \)\(34\!\cdots\!00\)\( \nu^{15} - \)\(50\!\cdots\!00\)\( \nu^{14} - \)\(70\!\cdots\!80\)\( \nu^{13} - \)\(10\!\cdots\!40\)\( \nu^{12} - \)\(88\!\cdots\!80\)\( \nu^{11} - \)\(12\!\cdots\!40\)\( \nu^{10} - \)\(67\!\cdots\!64\)\( \nu^{9} - \)\(88\!\cdots\!52\)\( \nu^{8} - \)\(31\!\cdots\!40\)\( \nu^{7} - \)\(36\!\cdots\!20\)\( \nu^{6} - \)\(83\!\cdots\!00\)\( \nu^{5} - \)\(76\!\cdots\!00\)\( \nu^{4} - \)\(11\!\cdots\!00\)\( \nu^{3} - \)\(57\!\cdots\!00\)\( \nu^{2} - \)\(59\!\cdots\!00\)\( \nu + \)\(12\!\cdots\!00\)\(\)\()/ \)\(84\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(16\!\cdots\!53\)\( \nu^{19} + \)\(10\!\cdots\!12\)\( \nu^{18} - \)\(15\!\cdots\!15\)\( \nu^{17} + \)\(82\!\cdots\!20\)\( \nu^{16} - \)\(60\!\cdots\!60\)\( \nu^{15} + \)\(26\!\cdots\!80\)\( \nu^{14} - \)\(12\!\cdots\!00\)\( \nu^{13} + \)\(43\!\cdots\!60\)\( \nu^{12} - \)\(15\!\cdots\!20\)\( \nu^{11} + \)\(38\!\cdots\!20\)\( \nu^{10} - \)\(10\!\cdots\!48\)\( \nu^{9} + \)\(16\!\cdots\!52\)\( \nu^{8} - \)\(42\!\cdots\!80\)\( \nu^{7} + \)\(22\!\cdots\!20\)\( \nu^{6} - \)\(65\!\cdots\!00\)\( \nu^{5} - \)\(57\!\cdots\!00\)\( \nu^{4} + \)\(33\!\cdots\!00\)\( \nu^{3} - \)\(10\!\cdots\!00\)\( \nu^{2} + \)\(10\!\cdots\!00\)\( \nu + \)\(53\!\cdots\!00\)\(\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(25\!\cdots\!29\)\( \nu^{19} - \)\(15\!\cdots\!24\)\( \nu^{18} + \)\(22\!\cdots\!15\)\( \nu^{17} - \)\(13\!\cdots\!00\)\( \nu^{16} + \)\(86\!\cdots\!60\)\( \nu^{15} - \)\(49\!\cdots\!00\)\( \nu^{14} + \)\(17\!\cdots\!20\)\( \nu^{13} - \)\(95\!\cdots\!80\)\( \nu^{12} + \)\(21\!\cdots\!40\)\( \nu^{11} - \)\(10\!\cdots\!80\)\( \nu^{10} + \)\(16\!\cdots\!84\)\( \nu^{9} - \)\(73\!\cdots\!64\)\( \nu^{8} + \)\(73\!\cdots\!40\)\( \nu^{7} - \)\(28\!\cdots\!40\)\( \nu^{6} + \)\(18\!\cdots\!00\)\( \nu^{5} - \)\(56\!\cdots\!00\)\( \nu^{4} + \)\(24\!\cdots\!00\)\( \nu^{3} - \)\(40\!\cdots\!00\)\( \nu^{2} + \)\(11\!\cdots\!00\)\( \nu + \)\(11\!\cdots\!00\)\(\)\()/ \)\(63\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/36\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 1519 \beta_{2} + 141 \beta_{1} - 117510611961387580904\)\()/1296\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{8} + 7 \beta_{7} + 227 \beta_{5} + 671787 \beta_{4} - 90503341 \beta_{3} - 16361058157166 \beta_{2} - 189993356209881425732 \beta_{1} + 3272193665213\)\()/46656\)
\(\nu^{4}\)\(=\)\((\)\(-13575 \beta_{16} + 29517 \beta_{15} - 17257 \beta_{13} + 26976 \beta_{12} - 227556 \beta_{11} - 1385528 \beta_{10} + 2058155 \beta_{9} + 81866204 \beta_{8} - 84058466325 \beta_{7} + 162598667649 \beta_{6} - 27337238790086 \beta_{5} + 144494037007035 \beta_{4} - 281198672344971877176 \beta_{3} + 278605710987654183234672 \beta_{2} - 25814672224335600045120 \beta_{1} + 22326235735894135855925494559722770113807\)\()/1679616\)
\(\nu^{5}\)\(=\)\((\)\(-2575287648 \beta_{19} - 13548022232 \beta_{18} - 5827627322860 \beta_{17} + 1453260582654 \beta_{16} + 7224069366228 \beta_{15} - 50420438188996 \beta_{14} - 1659694146574 \beta_{13} - 3294119686926 \beta_{12} - 17806451342152 \beta_{11} - 30116245912354 \beta_{10} + 63930164527513854 \beta_{9} - 22190540523448070521 \beta_{8} - 203549443232772264335 \beta_{7} - 472855645559796338 \beta_{6} - 828873915251826129931 \beta_{5} - 17319007876090264910699997 \beta_{4} + 2687385612605780775380247555 \beta_{3} + 485695741189351620476081801387248 \beta_{2} + 2665569763599380936657250234195015681232 \beta_{1} - 97138614224876300456438161121757\)\()/3779136\)
\(\nu^{6}\)\(=\)\((\)\(\)\(39\!\cdots\!73\)\( \beta_{16} - \)\(71\!\cdots\!75\)\( \beta_{15} + \)\(40\!\cdots\!15\)\( \beta_{13} - \)\(65\!\cdots\!96\)\( \beta_{12} + \)\(76\!\cdots\!28\)\( \beta_{11} + \)\(43\!\cdots\!52\)\( \beta_{10} - \)\(31\!\cdots\!01\)\( \beta_{9} - \)\(25\!\cdots\!84\)\( \beta_{8} + \)\(19\!\cdots\!11\)\( \beta_{7} - \)\(41\!\cdots\!55\)\( \beta_{6} + \)\(93\!\cdots\!26\)\( \beta_{5} - \)\(26\!\cdots\!93\)\( \beta_{4} + \)\(47\!\cdots\!60\)\( \beta_{3} - \)\(43\!\cdots\!40\)\( \beta_{2} + \)\(40\!\cdots\!60\)\( \beta_{1} - \)\(31\!\cdots\!01\)\(\)\()/ 136048896 \)
\(\nu^{7}\)\(=\)\((\)\(\)\(73\!\cdots\!60\)\( \beta_{19} + \)\(35\!\cdots\!40\)\( \beta_{18} + \)\(18\!\cdots\!84\)\( \beta_{17} - \)\(37\!\cdots\!02\)\( \beta_{16} - \)\(20\!\cdots\!96\)\( \beta_{15} + \)\(12\!\cdots\!12\)\( \beta_{14} + \)\(55\!\cdots\!86\)\( \beta_{13} + \)\(10\!\cdots\!78\)\( \beta_{12} + \)\(11\!\cdots\!28\)\( \beta_{11} + \)\(14\!\cdots\!74\)\( \beta_{10} - \)\(18\!\cdots\!50\)\( \beta_{9} + \)\(41\!\cdots\!39\)\( \beta_{8} + \)\(47\!\cdots\!73\)\( \beta_{7} + \)\(35\!\cdots\!50\)\( \beta_{6} - \)\(13\!\cdots\!47\)\( \beta_{5} + \)\(39\!\cdots\!39\)\( \beta_{4} - \)\(64\!\cdots\!85\)\( \beta_{3} - \)\(11\!\cdots\!44\)\( \beta_{2} - \)\(40\!\cdots\!00\)\( \beta_{1} + \)\(23\!\cdots\!03\)\(\)\()/ 306110016 \)
\(\nu^{8}\)\(=\)\((\)\(-\)\(28\!\cdots\!65\)\( \beta_{16} + \)\(46\!\cdots\!83\)\( \beta_{15} - \)\(25\!\cdots\!83\)\( \beta_{13} + \)\(42\!\cdots\!24\)\( \beta_{12} - \)\(60\!\cdots\!92\)\( \beta_{11} - \)\(33\!\cdots\!60\)\( \beta_{10} + \)\(28\!\cdots\!13\)\( \beta_{9} + \)\(19\!\cdots\!96\)\( \beta_{8} - \)\(10\!\cdots\!91\)\( \beta_{7} + \)\(24\!\cdots\!11\)\( \beta_{6} - \)\(75\!\cdots\!42\)\( \beta_{5} + \)\(93\!\cdots\!57\)\( \beta_{4} - \)\(26\!\cdots\!36\)\( \beta_{3} + \)\(29\!\cdots\!72\)\( \beta_{2} - \)\(27\!\cdots\!44\)\( \beta_{1} + \)\(15\!\cdots\!37\)\(\)\()/ 3673320192 \)
\(\nu^{9}\)\(=\)\((\)\(-\)\(51\!\cdots\!52\)\( \beta_{19} - \)\(21\!\cdots\!88\)\( \beta_{18} - \)\(14\!\cdots\!88\)\( \beta_{17} + \)\(23\!\cdots\!10\)\( \beta_{16} + \)\(14\!\cdots\!44\)\( \beta_{15} - \)\(89\!\cdots\!68\)\( \beta_{14} - \)\(44\!\cdots\!58\)\( \beta_{13} - \)\(75\!\cdots\!10\)\( \beta_{12} - \)\(12\!\cdots\!04\)\( \beta_{11} - \)\(15\!\cdots\!54\)\( \beta_{10} + \)\(13\!\cdots\!06\)\( \beta_{9} - \)\(24\!\cdots\!07\)\( \beta_{8} - \)\(32\!\cdots\!53\)\( \beta_{7} - \)\(40\!\cdots\!22\)\( \beta_{6} + \)\(18\!\cdots\!87\)\( \beta_{5} - \)\(27\!\cdots\!87\)\( \beta_{4} + \)\(45\!\cdots\!89\)\( \beta_{3} + \)\(81\!\cdots\!32\)\( \beta_{2} + \)\(21\!\cdots\!88\)\( \beta_{1} - \)\(16\!\cdots\!15\)\(\)\()/ 8264970432 \)
\(\nu^{10}\)\(=\)\((\)\(\)\(54\!\cdots\!27\)\( \beta_{16} - \)\(85\!\cdots\!33\)\( \beta_{15} + \)\(44\!\cdots\!93\)\( \beta_{13} - \)\(80\!\cdots\!08\)\( \beta_{12} + \)\(12\!\cdots\!40\)\( \beta_{11} + \)\(69\!\cdots\!44\)\( \beta_{10} - \)\(59\!\cdots\!63\)\( \beta_{9} - \)\(41\!\cdots\!52\)\( \beta_{8} + \)\(16\!\cdots\!57\)\( \beta_{7} - \)\(35\!\cdots\!01\)\( \beta_{6} + \)\(16\!\cdots\!02\)\( \beta_{5} - \)\(29\!\cdots\!23\)\( \beta_{4} + \)\(44\!\cdots\!20\)\( \beta_{3} - \)\(61\!\cdots\!60\)\( \beta_{2} + \)\(56\!\cdots\!52\)\( \beta_{1} - \)\(25\!\cdots\!99\)\(\)\()/ 297538935552 \)
\(\nu^{11}\)\(=\)\((\)\(\)\(97\!\cdots\!72\)\( \beta_{19} + \)\(32\!\cdots\!48\)\( \beta_{18} + \)\(29\!\cdots\!20\)\( \beta_{17} - \)\(42\!\cdots\!66\)\( \beta_{16} - \)\(27\!\cdots\!72\)\( \beta_{15} + \)\(17\!\cdots\!64\)\( \beta_{14} + \)\(94\!\cdots\!66\)\( \beta_{13} + \)\(15\!\cdots\!34\)\( \beta_{12} + \)\(33\!\cdots\!88\)\( \beta_{11} + \)\(38\!\cdots\!86\)\( \beta_{10} - \)\(25\!\cdots\!66\)\( \beta_{9} + \)\(42\!\cdots\!29\)\( \beta_{8} + \)\(63\!\cdots\!31\)\( \beta_{7} + \)\(11\!\cdots\!62\)\( \beta_{6} - \)\(50\!\cdots\!97\)\( \beta_{5} + \)\(55\!\cdots\!01\)\( \beta_{4} - \)\(89\!\cdots\!27\)\( \beta_{3} - \)\(16\!\cdots\!08\)\( \beta_{2} - \)\(36\!\cdots\!28\)\( \beta_{1} + \)\(32\!\cdots\!41\)\(\)\()/ 669462604992 \)
\(\nu^{12}\)\(=\)\((\)\(-\)\(33\!\cdots\!39\)\( \beta_{16} + \)\(51\!\cdots\!65\)\( \beta_{15} - \)\(25\!\cdots\!85\)\( \beta_{13} + \)\(50\!\cdots\!28\)\( \beta_{12} - \)\(81\!\cdots\!80\)\( \beta_{11} - \)\(45\!\cdots\!92\)\( \beta_{10} + \)\(37\!\cdots\!19\)\( \beta_{9} + \)\(26\!\cdots\!92\)\( \beta_{8} - \)\(74\!\cdots\!45\)\( \beta_{7} + \)\(13\!\cdots\!25\)\( \beta_{6} - \)\(10\!\cdots\!74\)\( \beta_{5} - \)\(64\!\cdots\!97\)\( \beta_{4} - \)\(25\!\cdots\!64\)\( \beta_{3} + \)\(42\!\cdots\!08\)\( \beta_{2} - \)\(39\!\cdots\!28\)\( \beta_{1} + \)\(14\!\cdots\!11\)\(\)\()/ 8033551259904 \)
\(\nu^{13}\)\(=\)\((\)\(-\)\(58\!\cdots\!80\)\( \beta_{19} - \)\(14\!\cdots\!60\)\( \beta_{18} - \)\(19\!\cdots\!08\)\( \beta_{17} + \)\(23\!\cdots\!54\)\( \beta_{16} + \)\(16\!\cdots\!92\)\( \beta_{15} - \)\(11\!\cdots\!04\)\( \beta_{14} - \)\(62\!\cdots\!82\)\( \beta_{13} - \)\(98\!\cdots\!46\)\( \beta_{12} - \)\(26\!\cdots\!16\)\( \beta_{11} - \)\(30\!\cdots\!98\)\( \beta_{10} + \)\(15\!\cdots\!90\)\( \beta_{9} - \)\(24\!\cdots\!73\)\( \beta_{8} - \)\(39\!\cdots\!95\)\( \beta_{7} - \)\(88\!\cdots\!30\)\( \beta_{6} + \)\(40\!\cdots\!73\)\( \beta_{5} - \)\(36\!\cdots\!85\)\( \beta_{4} + \)\(56\!\cdots\!23\)\( \beta_{3} + \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(20\!\cdots\!40\)\( \beta_{1} - \)\(20\!\cdots\!73\)\(\)\()/ 18075490334784 \)
\(\nu^{14}\)\(=\)\((\)\(\)\(22\!\cdots\!75\)\( \beta_{16} - \)\(34\!\cdots\!69\)\( \beta_{15} + \)\(16\!\cdots\!69\)\( \beta_{13} - \)\(34\!\cdots\!92\)\( \beta_{12} + \)\(56\!\cdots\!52\)\( \beta_{11} + \)\(32\!\cdots\!16\)\( \beta_{10} - \)\(24\!\cdots\!35\)\( \beta_{9} - \)\(18\!\cdots\!08\)\( \beta_{8} + \)\(32\!\cdots\!25\)\( \beta_{7} - \)\(42\!\cdots\!13\)\( \beta_{6} + \)\(74\!\cdots\!42\)\( \beta_{5} + \)\(94\!\cdots\!05\)\( \beta_{4} + \)\(15\!\cdots\!12\)\( \beta_{3} - \)\(32\!\cdots\!24\)\( \beta_{2} + \)\(29\!\cdots\!80\)\( \beta_{1} - \)\(88\!\cdots\!99\)\(\)\()/ 24100653779712 \)
\(\nu^{15}\)\(=\)\((\)\(\)\(12\!\cdots\!12\)\( \beta_{19} + \)\(18\!\cdots\!28\)\( \beta_{18} + \)\(44\!\cdots\!00\)\( \beta_{17} - \)\(49\!\cdots\!06\)\( \beta_{16} - \)\(37\!\cdots\!12\)\( \beta_{15} + \)\(27\!\cdots\!24\)\( \beta_{14} + \)\(14\!\cdots\!06\)\( \beta_{13} + \)\(22\!\cdots\!34\)\( \beta_{12} + \)\(73\!\cdots\!28\)\( \beta_{11} + \)\(80\!\cdots\!26\)\( \beta_{10} - \)\(34\!\cdots\!06\)\( \beta_{9} + \)\(51\!\cdots\!19\)\( \beta_{8} + \)\(89\!\cdots\!25\)\( \beta_{7} + \)\(24\!\cdots\!82\)\( \beta_{6} - \)\(10\!\cdots\!51\)\( \beta_{5} + \)\(85\!\cdots\!43\)\( \beta_{4} - \)\(12\!\cdots\!25\)\( \beta_{3} - \)\(23\!\cdots\!12\)\( \beta_{2} - \)\(42\!\cdots\!48\)\( \beta_{1} + \)\(46\!\cdots\!23\)\(\)\()/ 18075490334784 \)
\(\nu^{16}\)\(=\)\((\)\(-\)\(51\!\cdots\!81\)\( \beta_{16} + \)\(76\!\cdots\!35\)\( \beta_{15} - \)\(33\!\cdots\!15\)\( \beta_{13} + \)\(79\!\cdots\!92\)\( \beta_{12} - \)\(12\!\cdots\!16\)\( \beta_{11} - \)\(75\!\cdots\!04\)\( \beta_{10} + \)\(54\!\cdots\!17\)\( \beta_{9} + \)\(42\!\cdots\!68\)\( \beta_{8} - \)\(36\!\cdots\!07\)\( \beta_{7} - \)\(16\!\cdots\!45\)\( \beta_{6} - \)\(17\!\cdots\!22\)\( \beta_{5} - \)\(31\!\cdots\!99\)\( \beta_{4} - \)\(33\!\cdots\!80\)\( \beta_{3} + \)\(78\!\cdots\!00\)\( \beta_{2} - \)\(73\!\cdots\!80\)\( \beta_{1} + \)\(18\!\cdots\!17\)\(\)\()/ 24100653779712 \)
\(\nu^{17}\)\(=\)\((\)\(-\)\(92\!\cdots\!40\)\( \beta_{19} - \)\(47\!\cdots\!60\)\( \beta_{18} - \)\(34\!\cdots\!52\)\( \beta_{17} + \)\(33\!\cdots\!86\)\( \beta_{16} + \)\(27\!\cdots\!48\)\( \beta_{15} - \)\(21\!\cdots\!56\)\( \beta_{14} - \)\(11\!\cdots\!38\)\( \beta_{13} - \)\(17\!\cdots\!54\)\( \beta_{12} - \)\(63\!\cdots\!24\)\( \beta_{11} - \)\(68\!\cdots\!02\)\( \beta_{10} + \)\(25\!\cdots\!30\)\( \beta_{9} - \)\(36\!\cdots\!87\)\( \beta_{8} - \)\(66\!\cdots\!89\)\( \beta_{7} - \)\(21\!\cdots\!10\)\( \beta_{6} + \)\(94\!\cdots\!91\)\( \beta_{5} - \)\(66\!\cdots\!67\)\( \beta_{4} + \)\(97\!\cdots\!45\)\( \beta_{3} + \)\(17\!\cdots\!12\)\( \beta_{2} + \)\(29\!\cdots\!40\)\( \beta_{1} - \)\(35\!\cdots\!59\)\(\)\()/ 6025163444928 \)
\(\nu^{18}\)\(=\)\((\)\(\)\(37\!\cdots\!85\)\( \beta_{16} - \)\(56\!\cdots\!11\)\( \beta_{15} + \)\(23\!\cdots\!11\)\( \beta_{13} - \)\(60\!\cdots\!68\)\( \beta_{12} + \)\(95\!\cdots\!24\)\( \beta_{11} + \)\(57\!\cdots\!20\)\( \beta_{10} - \)\(38\!\cdots\!61\)\( \beta_{9} - \)\(31\!\cdots\!72\)\( \beta_{8} + \)\(22\!\cdots\!67\)\( \beta_{7} + \)\(86\!\cdots\!13\)\( \beta_{6} + \)\(12\!\cdots\!54\)\( \beta_{5} + \)\(30\!\cdots\!31\)\( \beta_{4} + \)\(23\!\cdots\!32\)\( \beta_{3} - \)\(63\!\cdots\!64\)\( \beta_{2} + \)\(59\!\cdots\!88\)\( \beta_{1} - \)\(12\!\cdots\!29\)\(\)\()/ 8033551259904 \)
\(\nu^{19}\)\(=\)\((\)\(\)\(59\!\cdots\!84\)\( \beta_{19} - \)\(24\!\cdots\!04\)\( \beta_{18} + \)\(23\!\cdots\!16\)\( \beta_{17} - \)\(20\!\cdots\!30\)\( \beta_{16} - \)\(18\!\cdots\!28\)\( \beta_{15} + \)\(15\!\cdots\!16\)\( \beta_{14} + \)\(79\!\cdots\!66\)\( \beta_{13} + \)\(11\!\cdots\!10\)\( \beta_{12} + \)\(48\!\cdots\!08\)\( \beta_{11} + \)\(51\!\cdots\!38\)\( \beta_{10} - \)\(16\!\cdots\!02\)\( \beta_{9} + \)\(23\!\cdots\!39\)\( \beta_{8} + \)\(43\!\cdots\!61\)\( \beta_{7} + \)\(15\!\cdots\!74\)\( \beta_{6} - \)\(70\!\cdots\!59\)\( \beta_{5} + \)\(46\!\cdots\!59\)\( \beta_{4} - \)\(65\!\cdots\!53\)\( \beta_{3} - \)\(11\!\cdots\!84\)\( \beta_{2} - \)\(18\!\cdots\!96\)\( \beta_{1} + \)\(23\!\cdots\!55\)\(\)\()/ 18075490334784 \)

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
4.67878e8i
4.43955e8i
3.58413e8i
3.39530e8i
3.00969e8i
2.56585e8i
2.39059e8i
1.32287e8i
1.26111e8i
96378.2i
96378.2i
1.26111e8i
1.32287e8i
2.39059e8i
2.56585e8i
3.00969e8i
3.39530e8i
3.58413e8i
4.43955e8i
4.67878e8i
1.68436e10i −3.79959e15 + 4.05786e15i −2.09920e20 9.39164e22i 6.83490e25 + 6.39988e25i −5.61847e27 2.29297e30i −2.02937e30 3.08364e31i 1.58189e33
2.2 1.59824e10i 5.28212e15 1.73274e15i −1.81649e20 8.56960e22i −2.76933e25 8.44208e25i 1.24551e28 1.72390e30i 2.48984e31 1.83050e31i −1.36963e33
2.3 1.29029e10i −3.61968e15 4.21913e15i −9.26974e19 1.27679e23i −5.44390e25 + 4.67043e25i 1.78063e27 2.43999e29i −4.69903e30 + 3.05438e31i −1.64743e33
2.4 1.22231e10i 1.27232e15 5.41150e15i −7.56171e19 1.11898e23i −6.61453e25 1.55517e25i −1.34063e28 2.23699e28i −2.76656e31 1.37703e31i 1.36774e33
2.5 1.08349e10i 3.60729e15 + 4.22973e15i −4.36076e19 5.11119e22i 4.58286e25 3.90845e25i −1.55423e27 3.26990e29i −4.87811e30 + 3.05157e31i 5.53791e32
2.6 9.23704e9i −3.04579e15 + 4.65041e15i −1.15360e19 1.92130e23i 4.29561e25 + 2.81341e25i 2.12741e26 5.75015e29i −1.23495e31 2.83283e31i −1.77471e33
2.7 8.60612e9i −5.53379e15 5.29477e14i −2.78318e17 1.34902e23i −4.55674e24 + 4.76244e25i 8.41299e27 6.32624e29i 3.03425e31 + 5.86002e30i 1.16098e33
2.8 4.76234e9i 4.09693e15 3.75743e15i 5.11071e19 1.11626e23i −1.78942e25 1.95110e25i 1.00296e28 5.94788e29i 2.66653e30 3.07879e31i 5.31601e32
2.9 4.54000e9i 5.46530e15 1.01669e15i 5.31753e19 1.75307e23i −4.61577e24 2.48125e25i −8.19169e27 5.76409e29i 2.88358e31 1.11130e31i −7.95893e32
2.10 3.46962e6i −1.28962e15 5.40740e15i 7.37870e19 7.05043e22i −1.87616e22 + 4.47450e21i 2.25837e27 5.12025e26i −2.75769e31 + 1.39470e31i −2.44623e29
2.11 3.46962e6i −1.28962e15 + 5.40740e15i 7.37870e19 7.05043e22i −1.87616e22 4.47450e21i 2.25837e27 5.12025e26i −2.75769e31 1.39470e31i −2.44623e29
2.12 4.54000e9i 5.46530e15 + 1.01669e15i 5.31753e19 1.75307e23i −4.61577e24 + 2.48125e25i −8.19169e27 5.76409e29i 2.88358e31 + 1.11130e31i −7.95893e32
2.13 4.76234e9i 4.09693e15 + 3.75743e15i 5.11071e19 1.11626e23i −1.78942e25 + 1.95110e25i 1.00296e28 5.94788e29i 2.66653e30 + 3.07879e31i 5.31601e32
2.14 8.60612e9i −5.53379e15 + 5.29477e14i −2.78318e17 1.34902e23i −4.55674e24 4.76244e25i 8.41299e27 6.32624e29i 3.03425e31 5.86002e30i 1.16098e33
2.15 9.23704e9i −3.04579e15 4.65041e15i −1.15360e19 1.92130e23i 4.29561e25 2.81341e25i 2.12741e26 5.75015e29i −1.23495e31 + 2.83283e31i −1.77471e33
2.16 1.08349e10i 3.60729e15 4.22973e15i −4.36076e19 5.11119e22i 4.58286e25 + 3.90845e25i −1.55423e27 3.26990e29i −4.87811e30 3.05157e31i 5.53791e32
2.17 1.22231e10i 1.27232e15 + 5.41150e15i −7.56171e19 1.11898e23i −6.61453e25 + 1.55517e25i −1.34063e28 2.23699e28i −2.76656e31 + 1.37703e31i 1.36774e33
2.18 1.29029e10i −3.61968e15 + 4.21913e15i −9.26974e19 1.27679e23i −5.44390e25 4.67043e25i 1.78063e27 2.43999e29i −4.69903e30 3.05438e31i −1.64743e33
2.19 1.59824e10i 5.28212e15 + 1.73274e15i −1.81649e20 8.56960e22i −2.76933e25 + 8.44208e25i 1.24551e28 1.72390e30i 2.48984e31 + 1.83050e31i −1.36963e33
2.20 1.68436e10i −3.79959e15 4.05786e15i −2.09920e20 9.39164e22i 6.83490e25 6.39988e25i −5.61847e27 2.29297e30i −2.02937e30 + 3.08364e31i 1.58189e33
\(n\): 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 3.67.b.b 20
3.b odd 2 1 inner 3.67.b.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.67.b.b 20 1.a even 1 1 trivial
3.67.b.b 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} + \cdots\) acting on \(S_{67}^{\mathrm{new}}(3, [\chi])\).