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 \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(82.7604085389\)
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 \( 2^{216}\cdot 3^{291}\cdot 5^{20}\cdot 7^{10}\cdot 11^{6}\cdot 13^{4} \)
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.

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

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

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