Properties

Label 1.112.a.a
Level 1
Weight 112
Character orbit 1.a
Self dual Yes
Analytic conductor 78.026
Analytic rank 0
Dimension 9
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 112 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(78.0257547452\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{135}\cdot 3^{56}\cdot 5^{16}\cdot 7^{7}\cdot 11^{3}\cdot 13\cdot 19\cdot 37^{3} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(811166749386264 + \beta_{1}) q^{2} +(\)\(25\!\cdots\!28\)\( - 625113423 \beta_{1} + \beta_{2}) q^{3} +(\)\(12\!\cdots\!28\)\( + 9945804401529458 \beta_{1} + 439252 \beta_{2} + \beta_{3}) q^{4} +(\)\(90\!\cdots\!70\)\( + \)\(20\!\cdots\!84\)\( \beta_{1} - 192373936543 \beta_{2} + 62042 \beta_{3} - \beta_{4}) q^{5} +(-\)\(23\!\cdots\!48\)\( + \)\(36\!\cdots\!33\)\( \beta_{1} + 12660797663077024 \beta_{2} - 1776098495 \beta_{3} + 1982 \beta_{4} - \beta_{5}) q^{6} +(\)\(87\!\cdots\!84\)\( - \)\(22\!\cdots\!76\)\( \beta_{1} + 40728023053536063228 \beta_{2} + 2786172113908 \beta_{3} - 18984915 \beta_{4} - 544 \beta_{5} - \beta_{6}) q^{7} +(\)\(36\!\cdots\!20\)\( + \)\(16\!\cdots\!81\)\( \beta_{1} - \)\(82\!\cdots\!40\)\( \beta_{2} + 7426322372361145 \beta_{3} + 36618818906 \beta_{4} + 1158411 \beta_{5} + 544 \beta_{6} + \beta_{7}) q^{8} +(\)\(49\!\cdots\!57\)\( + \)\(39\!\cdots\!76\)\( \beta_{1} + \)\(43\!\cdots\!80\)\( \beta_{2} - 3785148141483057164 \beta_{3} + 7815149744392 \beta_{4} - 432334485 \beta_{5} - 106965 \beta_{6} + 2 \beta_{7} + \beta_{8}) q^{9} +O(q^{10})\) \( q +(811166749386264 + \beta_{1}) q^{2} +(\)\(25\!\cdots\!28\)\( - 625113423 \beta_{1} + \beta_{2}) q^{3} +(\)\(12\!\cdots\!28\)\( + 9945804401529458 \beta_{1} + 439252 \beta_{2} + \beta_{3}) q^{4} +(\)\(90\!\cdots\!70\)\( + \)\(20\!\cdots\!84\)\( \beta_{1} - 192373936543 \beta_{2} + 62042 \beta_{3} - \beta_{4}) q^{5} +(-\)\(23\!\cdots\!48\)\( + \)\(36\!\cdots\!33\)\( \beta_{1} + 12660797663077024 \beta_{2} - 1776098495 \beta_{3} + 1982 \beta_{4} - \beta_{5}) q^{6} +(\)\(87\!\cdots\!84\)\( - \)\(22\!\cdots\!76\)\( \beta_{1} + 40728023053536063228 \beta_{2} + 2786172113908 \beta_{3} - 18984915 \beta_{4} - 544 \beta_{5} - \beta_{6}) q^{7} +(\)\(36\!\cdots\!20\)\( + \)\(16\!\cdots\!81\)\( \beta_{1} - \)\(82\!\cdots\!40\)\( \beta_{2} + 7426322372361145 \beta_{3} + 36618818906 \beta_{4} + 1158411 \beta_{5} + 544 \beta_{6} + \beta_{7}) q^{8} +(\)\(49\!\cdots\!57\)\( + \)\(39\!\cdots\!76\)\( \beta_{1} + \)\(43\!\cdots\!80\)\( \beta_{2} - 3785148141483057164 \beta_{3} + 7815149744392 \beta_{4} - 432334485 \beta_{5} - 106965 \beta_{6} + 2 \beta_{7} + \beta_{8}) q^{9} +(\)\(78\!\cdots\!20\)\( + \)\(28\!\cdots\!62\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(12\!\cdots\!64\)\( \beta_{3} + 5593656628680360 \beta_{4} + 156983200100 \beta_{5} + 46704872 \beta_{6} + 169680 \beta_{7} + 24 \beta_{8}) q^{10} +(\)\(10\!\cdots\!12\)\( + \)\(38\!\cdots\!91\)\( \beta_{1} - \)\(18\!\cdots\!65\)\( \beta_{2} + \)\(41\!\cdots\!32\)\( \beta_{3} + 168361756345494210 \beta_{4} + 12201234936324 \beta_{5} - 20487424510 \beta_{6} + 30257048 \beta_{7} - 26676 \beta_{8}) q^{11} +(\)\(69\!\cdots\!84\)\( - \)\(53\!\cdots\!88\)\( \beta_{1} + \)\(96\!\cdots\!76\)\( \beta_{2} - \)\(37\!\cdots\!00\)\( \beta_{3} - \)\(27\!\cdots\!44\)\( \beta_{4} - 20448558301736304 \beta_{5} - 11194670097216 \beta_{6} - 4202724624 \beta_{7} + 3451200 \beta_{8}) q^{12} +(-\)\(11\!\cdots\!82\)\( + \)\(26\!\cdots\!28\)\( \beta_{1} + \)\(94\!\cdots\!41\)\( \beta_{2} - \)\(19\!\cdots\!78\)\( \beta_{3} + \)\(55\!\cdots\!19\)\( \beta_{4} - 2255854112058309782 \beta_{5} + 53881994894122 \beta_{6} + 165977519484 \beta_{7} - 254870850 \beta_{8}) q^{13} +(-\)\(77\!\cdots\!44\)\( + \)\(17\!\cdots\!70\)\( \beta_{1} + \)\(19\!\cdots\!40\)\( \beta_{2} - \)\(50\!\cdots\!94\)\( \beta_{3} + \)\(50\!\cdots\!68\)\( \beta_{4} - \)\(21\!\cdots\!06\)\( \beta_{5} + 32386932686384480 \beta_{6} + 939225511616 \beta_{7} + 13262786208 \beta_{8}) q^{14} +(-\)\(29\!\cdots\!60\)\( - \)\(52\!\cdots\!36\)\( \beta_{1} + \)\(68\!\cdots\!40\)\( \beta_{2} + \)\(33\!\cdots\!88\)\( \beta_{3} - \)\(13\!\cdots\!25\)\( \beta_{4} - \)\(38\!\cdots\!00\)\( \beta_{5} - 1156059159208777071 \beta_{6} - 382957011300240 \beta_{7} - 531302463432 \beta_{8}) q^{15} +(\)\(31\!\cdots\!36\)\( + \)\(46\!\cdots\!28\)\( \beta_{1} - \)\(57\!\cdots\!16\)\( \beta_{2} + \)\(11\!\cdots\!08\)\( \beta_{3} - \)\(77\!\cdots\!44\)\( \beta_{4} + \)\(29\!\cdots\!16\)\( \beta_{5} + 2332059480839184640 \beta_{6} + 20463717115894488 \beta_{7} + 17214333970944 \beta_{8}) q^{16} +(-\)\(24\!\cdots\!26\)\( - \)\(71\!\cdots\!92\)\( \beta_{1} - \)\(76\!\cdots\!44\)\( \beta_{2} + \)\(17\!\cdots\!72\)\( \beta_{3} - \)\(27\!\cdots\!80\)\( \beta_{4} + \)\(13\!\cdots\!59\)\( \beta_{5} + \)\(79\!\cdots\!11\)\( \beta_{6} - 668532521885043270 \beta_{7} - 465580026922275 \beta_{8}) q^{17} +(\)\(15\!\cdots\!48\)\( + \)\(43\!\cdots\!85\)\( \beta_{1} + \)\(24\!\cdots\!48\)\( \beta_{2} + \)\(29\!\cdots\!68\)\( \beta_{3} - \)\(14\!\cdots\!40\)\( \beta_{4} - \)\(18\!\cdots\!24\)\( \beta_{5} - \)\(23\!\cdots\!96\)\( \beta_{6} + 15768547504982949600 \beta_{7} + 10743057227494800 \beta_{8}) q^{18} +(-\)\(94\!\cdots\!80\)\( - \)\(61\!\cdots\!75\)\( \beta_{1} - \)\(96\!\cdots\!47\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(77\!\cdots\!10\)\( \beta_{4} + \)\(17\!\cdots\!52\)\( \beta_{5} + \)\(27\!\cdots\!70\)\( \beta_{6} - \)\(28\!\cdots\!16\)\( \beta_{7} - 214854891759298908 \beta_{8}) q^{19} +(\)\(86\!\cdots\!60\)\( + \)\(87\!\cdots\!32\)\( \beta_{1} - \)\(11\!\cdots\!24\)\( \beta_{2} + \)\(45\!\cdots\!46\)\( \beta_{3} + \)\(12\!\cdots\!32\)\( \beta_{4} + \)\(36\!\cdots\!00\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6} + \)\(38\!\cdots\!00\)\( \beta_{7} + 3768188694918186240 \beta_{8}) q^{20} +(\)\(59\!\cdots\!12\)\( + \)\(57\!\cdots\!96\)\( \beta_{1} + \)\(15\!\cdots\!64\)\( \beta_{2} + \)\(78\!\cdots\!44\)\( \beta_{3} - \)\(18\!\cdots\!64\)\( \beta_{4} - \)\(90\!\cdots\!82\)\( \beta_{5} - \)\(99\!\cdots\!90\)\( \beta_{6} - \)\(37\!\cdots\!08\)\( \beta_{7} - 58463857513820712954 \beta_{8}) q^{21} +(\)\(14\!\cdots\!68\)\( + \)\(16\!\cdots\!63\)\( \beta_{1} - \)\(83\!\cdots\!04\)\( \beta_{2} + \)\(85\!\cdots\!67\)\( \beta_{3} - \)\(17\!\cdots\!66\)\( \beta_{4} + \)\(49\!\cdots\!73\)\( \beta_{5} + \)\(18\!\cdots\!92\)\( \beta_{6} + \)\(18\!\cdots\!24\)\( \beta_{7} + \)\(80\!\cdots\!00\)\( \beta_{8}) q^{22} +(-\)\(47\!\cdots\!92\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(76\!\cdots\!24\)\( \beta_{2} - \)\(81\!\cdots\!60\)\( \beta_{3} + \)\(20\!\cdots\!15\)\( \beta_{4} + \)\(70\!\cdots\!20\)\( \beta_{5} - \)\(18\!\cdots\!95\)\( \beta_{6} + \)\(18\!\cdots\!40\)\( \beta_{7} - \)\(99\!\cdots\!00\)\( \beta_{8}) q^{23} +(-\)\(14\!\cdots\!40\)\( - \)\(16\!\cdots\!32\)\( \beta_{1} + \)\(45\!\cdots\!80\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(24\!\cdots\!52\)\( \beta_{4} - \)\(13\!\cdots\!32\)\( \beta_{5} + \)\(80\!\cdots\!40\)\( \beta_{6} - \)\(62\!\cdots\!52\)\( \beta_{7} + \)\(11\!\cdots\!24\)\( \beta_{8}) q^{24} +(-\)\(50\!\cdots\!25\)\( + \)\(61\!\cdots\!40\)\( \beta_{1} + \)\(16\!\cdots\!80\)\( \beta_{2} - \)\(27\!\cdots\!60\)\( \beta_{3} - \)\(27\!\cdots\!40\)\( \beta_{4} + \)\(70\!\cdots\!50\)\( \beta_{5} + \)\(81\!\cdots\!30\)\( \beta_{6} + \)\(96\!\cdots\!00\)\( \beta_{7} - \)\(10\!\cdots\!90\)\( \beta_{8}) q^{25} +(\)\(10\!\cdots\!12\)\( - \)\(14\!\cdots\!10\)\( \beta_{1} + \)\(76\!\cdots\!56\)\( \beta_{2} + \)\(42\!\cdots\!08\)\( \beta_{3} - \)\(15\!\cdots\!36\)\( \beta_{4} + \)\(30\!\cdots\!96\)\( \beta_{5} - \)\(19\!\cdots\!20\)\( \beta_{6} - \)\(10\!\cdots\!24\)\( \beta_{7} + \)\(97\!\cdots\!88\)\( \beta_{8}) q^{26} +(\)\(40\!\cdots\!80\)\( + \)\(13\!\cdots\!94\)\( \beta_{1} + \)\(73\!\cdots\!90\)\( \beta_{2} + \)\(40\!\cdots\!40\)\( \beta_{3} + \)\(29\!\cdots\!34\)\( \beta_{4} - \)\(61\!\cdots\!76\)\( \beta_{5} + \)\(20\!\cdots\!46\)\( \beta_{6} + \)\(94\!\cdots\!64\)\( \beta_{7} - \)\(76\!\cdots\!00\)\( \beta_{8}) q^{27} +(\)\(43\!\cdots\!52\)\( - \)\(12\!\cdots\!28\)\( \beta_{1} + \)\(67\!\cdots\!44\)\( \beta_{2} + \)\(91\!\cdots\!20\)\( \beta_{3} - \)\(53\!\cdots\!16\)\( \beta_{4} + \)\(18\!\cdots\!84\)\( \beta_{5} - \)\(13\!\cdots\!64\)\( \beta_{6} - \)\(65\!\cdots\!36\)\( \beta_{7} + \)\(53\!\cdots\!00\)\( \beta_{8}) q^{28} +(-\)\(17\!\cdots\!70\)\( + \)\(98\!\cdots\!92\)\( \beta_{1} + \)\(22\!\cdots\!89\)\( \beta_{2} - \)\(16\!\cdots\!38\)\( \beta_{3} - \)\(10\!\cdots\!81\)\( \beta_{4} + \)\(24\!\cdots\!16\)\( \beta_{5} + \)\(43\!\cdots\!80\)\( \beta_{6} + \)\(33\!\cdots\!96\)\( \beta_{7} - \)\(31\!\cdots\!52\)\( \beta_{8}) q^{29} +(-\)\(20\!\cdots\!60\)\( - \)\(23\!\cdots\!62\)\( \beta_{1} + \)\(12\!\cdots\!84\)\( \beta_{2} - \)\(20\!\cdots\!86\)\( \beta_{3} + \)\(40\!\cdots\!88\)\( \beta_{4} - \)\(28\!\cdots\!50\)\( \beta_{5} + \)\(14\!\cdots\!80\)\( \beta_{6} - \)\(99\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!60\)\( \beta_{8}) q^{30} +(\)\(75\!\cdots\!32\)\( - \)\(51\!\cdots\!24\)\( \beta_{1} - \)\(46\!\cdots\!08\)\( \beta_{2} + \)\(87\!\cdots\!00\)\( \beta_{3} + \)\(26\!\cdots\!04\)\( \beta_{4} + \)\(84\!\cdots\!64\)\( \beta_{5} - \)\(29\!\cdots\!40\)\( \beta_{6} - \)\(29\!\cdots\!68\)\( \beta_{7} - \)\(37\!\cdots\!84\)\( \beta_{8}) q^{31} +(\)\(83\!\cdots\!44\)\( + \)\(25\!\cdots\!48\)\( \beta_{1} - \)\(92\!\cdots\!72\)\( \beta_{2} + \)\(79\!\cdots\!12\)\( \beta_{3} - \)\(15\!\cdots\!60\)\( \beta_{4} + \)\(54\!\cdots\!84\)\( \beta_{5} + \)\(20\!\cdots\!36\)\( \beta_{6} + \)\(67\!\cdots\!00\)\( \beta_{7} - \)\(15\!\cdots\!00\)\( \beta_{8}) q^{32} +(-\)\(34\!\cdots\!64\)\( - \)\(29\!\cdots\!64\)\( \beta_{1} - \)\(23\!\cdots\!88\)\( \beta_{2} - \)\(23\!\cdots\!92\)\( \beta_{3} - \)\(81\!\cdots\!96\)\( \beta_{4} - \)\(62\!\cdots\!15\)\( \beta_{5} - \)\(73\!\cdots\!35\)\( \beta_{6} - \)\(57\!\cdots\!26\)\( \beta_{7} + \)\(32\!\cdots\!75\)\( \beta_{8}) q^{33} +(-\)\(27\!\cdots\!84\)\( + \)\(17\!\cdots\!54\)\( \beta_{1} - \)\(82\!\cdots\!36\)\( \beta_{2} - \)\(16\!\cdots\!92\)\( \beta_{3} + \)\(73\!\cdots\!20\)\( \beta_{4} + \)\(20\!\cdots\!72\)\( \beta_{5} + \)\(31\!\cdots\!20\)\( \beta_{6} + \)\(33\!\cdots\!84\)\( \beta_{7} - \)\(29\!\cdots\!08\)\( \beta_{8}) q^{34} +(\)\(64\!\cdots\!20\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} + \)\(74\!\cdots\!40\)\( \beta_{2} - \)\(10\!\cdots\!56\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4} + \)\(43\!\cdots\!00\)\( \beta_{5} + \)\(12\!\cdots\!92\)\( \beta_{6} - \)\(14\!\cdots\!20\)\( \beta_{7} + \)\(20\!\cdots\!64\)\( \beta_{8}) q^{35} +(\)\(36\!\cdots\!96\)\( + \)\(18\!\cdots\!90\)\( \beta_{1} + \)\(49\!\cdots\!12\)\( \beta_{2} + \)\(54\!\cdots\!09\)\( \beta_{3} - \)\(16\!\cdots\!68\)\( \beta_{4} - \)\(59\!\cdots\!56\)\( \beta_{5} - \)\(69\!\cdots\!00\)\( \beta_{6} + \)\(41\!\cdots\!40\)\( \beta_{7} - \)\(11\!\cdots\!80\)\( \beta_{8}) q^{36} +(\)\(33\!\cdots\!54\)\( + \)\(63\!\cdots\!24\)\( \beta_{1} + \)\(10\!\cdots\!93\)\( \beta_{2} + \)\(33\!\cdots\!70\)\( \beta_{3} + \)\(24\!\cdots\!43\)\( \beta_{4} + \)\(12\!\cdots\!78\)\( \beta_{5} + \)\(50\!\cdots\!62\)\( \beta_{6} - \)\(53\!\cdots\!72\)\( \beta_{7} + \)\(53\!\cdots\!50\)\( \beta_{8}) q^{37} +(-\)\(23\!\cdots\!20\)\( + \)\(16\!\cdots\!37\)\( \beta_{1} - \)\(64\!\cdots\!32\)\( \beta_{2} - \)\(84\!\cdots\!59\)\( \beta_{3} + \)\(28\!\cdots\!26\)\( \beta_{4} + \)\(83\!\cdots\!83\)\( \beta_{5} + \)\(14\!\cdots\!32\)\( \beta_{6} - \)\(13\!\cdots\!24\)\( \beta_{7} - \)\(20\!\cdots\!00\)\( \beta_{8}) q^{38} +(\)\(36\!\cdots\!24\)\( + \)\(29\!\cdots\!40\)\( \beta_{1} - \)\(26\!\cdots\!64\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(88\!\cdots\!45\)\( \beta_{4} - \)\(59\!\cdots\!36\)\( \beta_{5} - \)\(10\!\cdots\!35\)\( \beta_{6} + \)\(20\!\cdots\!48\)\( \beta_{7} + \)\(60\!\cdots\!24\)\( \beta_{8}) q^{39} +(\)\(13\!\cdots\!00\)\( + \)\(15\!\cdots\!30\)\( \beta_{1} - \)\(12\!\cdots\!80\)\( \beta_{2} + \)\(15\!\cdots\!50\)\( \beta_{3} - \)\(30\!\cdots\!60\)\( \beta_{4} + \)\(93\!\cdots\!50\)\( \beta_{5} + \)\(38\!\cdots\!40\)\( \beta_{6} + \)\(96\!\cdots\!50\)\( \beta_{7} - \)\(92\!\cdots\!20\)\( \beta_{8}) q^{40} +(\)\(11\!\cdots\!42\)\( + \)\(32\!\cdots\!64\)\( \beta_{1} + \)\(28\!\cdots\!84\)\( \beta_{2} + \)\(37\!\cdots\!44\)\( \beta_{3} + \)\(12\!\cdots\!88\)\( \beta_{4} + \)\(51\!\cdots\!46\)\( \beta_{5} - \)\(64\!\cdots\!50\)\( \beta_{6} - \)\(99\!\cdots\!40\)\( \beta_{7} - \)\(28\!\cdots\!70\)\( \beta_{8}) q^{41} +(\)\(22\!\cdots\!68\)\( + \)\(29\!\cdots\!60\)\( \beta_{1} + \)\(23\!\cdots\!16\)\( \beta_{2} - \)\(15\!\cdots\!12\)\( \beta_{3} + \)\(58\!\cdots\!60\)\( \beta_{4} - \)\(24\!\cdots\!84\)\( \beta_{5} - \)\(12\!\cdots\!36\)\( \beta_{6} + \)\(58\!\cdots\!00\)\( \beta_{7} + \)\(31\!\cdots\!00\)\( \beta_{8}) q^{42} +(-\)\(21\!\cdots\!12\)\( + \)\(22\!\cdots\!47\)\( \beta_{1} + \)\(49\!\cdots\!75\)\( \beta_{2} - \)\(68\!\cdots\!96\)\( \beta_{3} - \)\(37\!\cdots\!24\)\( \beta_{4} - \)\(18\!\cdots\!36\)\( \beta_{5} - \)\(32\!\cdots\!44\)\( \beta_{6} - \)\(23\!\cdots\!84\)\( \beta_{7} - \)\(15\!\cdots\!00\)\( \beta_{8}) q^{43} +(\)\(61\!\cdots\!36\)\( + \)\(30\!\cdots\!52\)\( \beta_{1} - \)\(19\!\cdots\!04\)\( \beta_{2} + \)\(16\!\cdots\!92\)\( \beta_{3} + \)\(10\!\cdots\!64\)\( \beta_{4} + \)\(43\!\cdots\!84\)\( \beta_{5} + \)\(69\!\cdots\!60\)\( \beta_{6} + \)\(61\!\cdots\!32\)\( \beta_{7} + \)\(48\!\cdots\!16\)\( \beta_{8}) q^{44} +(\)\(17\!\cdots\!90\)\( + \)\(26\!\cdots\!48\)\( \beta_{1} - \)\(55\!\cdots\!91\)\( \beta_{2} + \)\(69\!\cdots\!34\)\( \beta_{3} + \)\(27\!\cdots\!63\)\( \beta_{4} - \)\(10\!\cdots\!50\)\( \beta_{5} - \)\(44\!\cdots\!10\)\( \beta_{6} - \)\(56\!\cdots\!00\)\( \beta_{7} - \)\(82\!\cdots\!70\)\( \beta_{8}) q^{45} +(-\)\(44\!\cdots\!28\)\( - \)\(80\!\cdots\!06\)\( \beta_{1} - \)\(20\!\cdots\!36\)\( \beta_{2} - \)\(98\!\cdots\!86\)\( \beta_{3} + \)\(12\!\cdots\!68\)\( \beta_{4} - \)\(22\!\cdots\!74\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6} - \)\(38\!\cdots\!00\)\( \beta_{7} - \)\(11\!\cdots\!00\)\( \beta_{8}) q^{46} +(\)\(16\!\cdots\!44\)\( - \)\(21\!\cdots\!44\)\( \beta_{1} + \)\(10\!\cdots\!20\)\( \beta_{2} - \)\(23\!\cdots\!32\)\( \beta_{3} - \)\(41\!\cdots\!58\)\( \beta_{4} + \)\(17\!\cdots\!88\)\( \beta_{5} - \)\(29\!\cdots\!98\)\( \beta_{6} + \)\(24\!\cdots\!72\)\( \beta_{7} + \)\(15\!\cdots\!00\)\( \beta_{8}) q^{47} +(-\)\(82\!\cdots\!92\)\( - \)\(44\!\cdots\!16\)\( \beta_{1} + \)\(37\!\cdots\!72\)\( \beta_{2} - \)\(23\!\cdots\!48\)\( \beta_{3} + \)\(18\!\cdots\!60\)\( \beta_{4} - \)\(23\!\cdots\!16\)\( \beta_{5} + \)\(15\!\cdots\!36\)\( \beta_{6} - \)\(75\!\cdots\!80\)\( \beta_{7} - \)\(66\!\cdots\!00\)\( \beta_{8}) q^{48} +(\)\(19\!\cdots\!93\)\( - \)\(55\!\cdots\!96\)\( \beta_{1} + \)\(79\!\cdots\!24\)\( \beta_{2} + \)\(55\!\cdots\!04\)\( \beta_{3} + \)\(43\!\cdots\!28\)\( \beta_{4} - \)\(76\!\cdots\!64\)\( \beta_{5} - \)\(40\!\cdots\!00\)\( \beta_{6} + \)\(11\!\cdots\!80\)\( \beta_{7} + \)\(15\!\cdots\!40\)\( \beta_{8}) q^{49} +(\)\(23\!\cdots\!00\)\( - \)\(12\!\cdots\!85\)\( \beta_{1} - \)\(18\!\cdots\!20\)\( \beta_{2} + \)\(26\!\cdots\!40\)\( \beta_{3} - \)\(43\!\cdots\!40\)\( \beta_{4} + \)\(15\!\cdots\!00\)\( \beta_{5} + \)\(60\!\cdots\!80\)\( \beta_{6} + \)\(88\!\cdots\!00\)\( \beta_{7} - \)\(89\!\cdots\!40\)\( \beta_{8}) q^{50} +(-\)\(95\!\cdots\!68\)\( - \)\(32\!\cdots\!10\)\( \beta_{1} - \)\(10\!\cdots\!58\)\( \beta_{2} - \)\(41\!\cdots\!40\)\( \beta_{3} - \)\(46\!\cdots\!70\)\( \beta_{4} + \)\(12\!\cdots\!08\)\( \beta_{5} - \)\(36\!\cdots\!70\)\( \beta_{6} - \)\(10\!\cdots\!24\)\( \beta_{7} - \)\(11\!\cdots\!12\)\( \beta_{8}) q^{51} +(-\)\(25\!\cdots\!96\)\( + \)\(15\!\cdots\!96\)\( \beta_{1} - \)\(75\!\cdots\!68\)\( \beta_{2} - \)\(44\!\cdots\!22\)\( \beta_{3} + \)\(10\!\cdots\!32\)\( \beta_{4} - \)\(67\!\cdots\!52\)\( \beta_{5} + \)\(86\!\cdots\!92\)\( \beta_{6} + \)\(24\!\cdots\!12\)\( \beta_{7} + \)\(58\!\cdots\!00\)\( \beta_{8}) q^{52} +(-\)\(36\!\cdots\!22\)\( + \)\(35\!\cdots\!28\)\( \beta_{1} + \)\(21\!\cdots\!93\)\( \beta_{2} + \)\(30\!\cdots\!90\)\( \beta_{3} + \)\(19\!\cdots\!11\)\( \beta_{4} + \)\(12\!\cdots\!06\)\( \beta_{5} + \)\(50\!\cdots\!74\)\( \beta_{6} - \)\(15\!\cdots\!44\)\( \beta_{7} - \)\(14\!\cdots\!50\)\( \beta_{8}) q^{53} +(\)\(50\!\cdots\!20\)\( + \)\(18\!\cdots\!38\)\( \beta_{1} + \)\(27\!\cdots\!80\)\( \beta_{2} + \)\(26\!\cdots\!54\)\( \beta_{3} - \)\(67\!\cdots\!76\)\( \beta_{4} - \)\(11\!\cdots\!86\)\( \beta_{5} - \)\(97\!\cdots\!40\)\( \beta_{6} - \)\(42\!\cdots\!48\)\( \beta_{7} + \)\(11\!\cdots\!76\)\( \beta_{8}) q^{54} +(\)\(60\!\cdots\!40\)\( + \)\(34\!\cdots\!08\)\( \beta_{1} + \)\(91\!\cdots\!84\)\( \beta_{2} + \)\(20\!\cdots\!04\)\( \beta_{3} - \)\(15\!\cdots\!87\)\( \beta_{4} + \)\(70\!\cdots\!00\)\( \beta_{5} + \)\(28\!\cdots\!75\)\( \beta_{6} - \)\(13\!\cdots\!00\)\( \beta_{7} + \)\(75\!\cdots\!00\)\( \beta_{8}) q^{55} +(-\)\(47\!\cdots\!20\)\( + \)\(23\!\cdots\!40\)\( \beta_{1} - \)\(39\!\cdots\!84\)\( \beta_{2} - \)\(23\!\cdots\!64\)\( \beta_{3} + \)\(58\!\cdots\!08\)\( \beta_{4} - \)\(45\!\cdots\!32\)\( \beta_{5} - \)\(25\!\cdots\!80\)\( \beta_{6} + \)\(16\!\cdots\!44\)\( \beta_{7} - \)\(39\!\cdots\!28\)\( \beta_{8}) q^{56} +(\)\(11\!\cdots\!60\)\( - \)\(26\!\cdots\!80\)\( \beta_{1} - \)\(49\!\cdots\!28\)\( \beta_{2} - \)\(12\!\cdots\!16\)\( \beta_{3} + \)\(43\!\cdots\!96\)\( \beta_{4} + \)\(12\!\cdots\!19\)\( \beta_{5} - \)\(60\!\cdots\!49\)\( \beta_{6} - \)\(46\!\cdots\!14\)\( \beta_{7} + \)\(89\!\cdots\!25\)\( \beta_{8}) q^{57} +(\)\(36\!\cdots\!20\)\( - \)\(60\!\cdots\!38\)\( \beta_{1} - \)\(37\!\cdots\!60\)\( \beta_{2} + \)\(68\!\cdots\!44\)\( \beta_{3} - \)\(15\!\cdots\!12\)\( \beta_{4} - \)\(11\!\cdots\!64\)\( \beta_{5} + \)\(10\!\cdots\!44\)\( \beta_{6} - \)\(10\!\cdots\!32\)\( \beta_{7} - \)\(39\!\cdots\!00\)\( \beta_{8}) q^{58} +(\)\(35\!\cdots\!60\)\( - \)\(90\!\cdots\!01\)\( \beta_{1} + \)\(94\!\cdots\!91\)\( \beta_{2} + \)\(36\!\cdots\!36\)\( \beta_{3} - \)\(99\!\cdots\!16\)\( \beta_{4} - \)\(20\!\cdots\!08\)\( \beta_{5} + \)\(54\!\cdots\!40\)\( \beta_{6} + \)\(53\!\cdots\!48\)\( \beta_{7} - \)\(48\!\cdots\!76\)\( \beta_{8}) q^{59} +(-\)\(14\!\cdots\!80\)\( - \)\(66\!\cdots\!68\)\( \beta_{1} + \)\(10\!\cdots\!60\)\( \beta_{2} - \)\(68\!\cdots\!76\)\( \beta_{3} + \)\(28\!\cdots\!80\)\( \beta_{4} - \)\(65\!\cdots\!00\)\( \beta_{5} - \)\(19\!\cdots\!28\)\( \beta_{6} - \)\(19\!\cdots\!20\)\( \beta_{7} + \)\(19\!\cdots\!24\)\( \beta_{8}) q^{60} +(-\)\(19\!\cdots\!38\)\( + \)\(11\!\cdots\!16\)\( \beta_{1} + \)\(10\!\cdots\!69\)\( \beta_{2} - \)\(10\!\cdots\!42\)\( \beta_{3} + \)\(19\!\cdots\!63\)\( \beta_{4} + \)\(27\!\cdots\!74\)\( \beta_{5} - \)\(16\!\cdots\!70\)\( \beta_{6} + \)\(24\!\cdots\!96\)\( \beta_{7} - \)\(33\!\cdots\!02\)\( \beta_{8}) q^{61} +(-\)\(19\!\cdots\!52\)\( + \)\(31\!\cdots\!80\)\( \beta_{1} - \)\(40\!\cdots\!68\)\( \beta_{2} - \)\(99\!\cdots\!32\)\( \beta_{3} + \)\(70\!\cdots\!80\)\( \beta_{4} + \)\(18\!\cdots\!96\)\( \beta_{5} + \)\(24\!\cdots\!84\)\( \beta_{6} + \)\(57\!\cdots\!20\)\( \beta_{7} - \)\(15\!\cdots\!00\)\( \beta_{8}) q^{62} +(\)\(14\!\cdots\!88\)\( + \)\(69\!\cdots\!84\)\( \beta_{1} - \)\(26\!\cdots\!68\)\( \beta_{2} + \)\(72\!\cdots\!60\)\( \beta_{3} - \)\(82\!\cdots\!43\)\( \beta_{4} - \)\(32\!\cdots\!68\)\( \beta_{5} - \)\(62\!\cdots\!97\)\( \beta_{6} - \)\(33\!\cdots\!28\)\( \beta_{7} + \)\(24\!\cdots\!00\)\( \beta_{8}) q^{63} +(\)\(15\!\cdots\!68\)\( + \)\(20\!\cdots\!44\)\( \beta_{1} - \)\(21\!\cdots\!28\)\( \beta_{2} + \)\(25\!\cdots\!20\)\( \beta_{3} + \)\(17\!\cdots\!16\)\( \beta_{4} + \)\(72\!\cdots\!12\)\( \beta_{5} + \)\(29\!\cdots\!00\)\( \beta_{6} + \)\(65\!\cdots\!00\)\( \beta_{7} - \)\(64\!\cdots\!00\)\( \beta_{8}) q^{64} +(-\)\(20\!\cdots\!60\)\( + \)\(12\!\cdots\!44\)\( \beta_{1} + \)\(30\!\cdots\!00\)\( \beta_{2} - \)\(46\!\cdots\!32\)\( \beta_{3} - \)\(25\!\cdots\!80\)\( \beta_{4} - \)\(23\!\cdots\!50\)\( \beta_{5} + \)\(22\!\cdots\!14\)\( \beta_{6} - \)\(12\!\cdots\!40\)\( \beta_{7} + \)\(52\!\cdots\!38\)\( \beta_{8}) q^{65} +(-\)\(11\!\cdots\!76\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} + \)\(18\!\cdots\!56\)\( \beta_{2} - \)\(17\!\cdots\!80\)\( \beta_{3} + \)\(18\!\cdots\!00\)\( \beta_{4} - \)\(66\!\cdots\!36\)\( \beta_{5} - \)\(59\!\cdots\!60\)\( \beta_{6} - \)\(31\!\cdots\!32\)\( \beta_{7} + \)\(18\!\cdots\!84\)\( \beta_{8}) q^{66} +(\)\(15\!\cdots\!24\)\( - \)\(42\!\cdots\!67\)\( \beta_{1} + \)\(34\!\cdots\!29\)\( \beta_{2} + \)\(16\!\cdots\!96\)\( \beta_{3} - \)\(23\!\cdots\!54\)\( \beta_{4} - \)\(34\!\cdots\!12\)\( \beta_{5} + \)\(52\!\cdots\!02\)\( \beta_{6} + \)\(74\!\cdots\!96\)\( \beta_{7} - \)\(73\!\cdots\!00\)\( \beta_{8}) q^{67} +(\)\(12\!\cdots\!72\)\( - \)\(56\!\cdots\!88\)\( \beta_{1} - \)\(37\!\cdots\!20\)\( \beta_{2} + \)\(62\!\cdots\!70\)\( \beta_{3} + \)\(43\!\cdots\!44\)\( \beta_{4} + \)\(11\!\cdots\!44\)\( \beta_{5} - \)\(26\!\cdots\!24\)\( \beta_{6} - \)\(45\!\cdots\!76\)\( \beta_{7} + \)\(97\!\cdots\!00\)\( \beta_{8}) q^{68} +(\)\(12\!\cdots\!44\)\( - \)\(72\!\cdots\!24\)\( \beta_{1} - \)\(17\!\cdots\!84\)\( \beta_{2} + \)\(67\!\cdots\!64\)\( \beta_{3} - \)\(11\!\cdots\!44\)\( \beta_{4} + \)\(11\!\cdots\!46\)\( \beta_{5} + \)\(24\!\cdots\!90\)\( \beta_{6} - \)\(13\!\cdots\!52\)\( \beta_{7} + \)\(79\!\cdots\!74\)\( \beta_{8}) q^{69} +(-\)\(40\!\cdots\!80\)\( + \)\(35\!\cdots\!44\)\( \beta_{1} - \)\(37\!\cdots\!68\)\( \beta_{2} - \)\(35\!\cdots\!88\)\( \beta_{3} + \)\(98\!\cdots\!24\)\( \beta_{4} - \)\(87\!\cdots\!00\)\( \beta_{5} - \)\(58\!\cdots\!40\)\( \beta_{6} + \)\(24\!\cdots\!00\)\( \beta_{7} - \)\(53\!\cdots\!80\)\( \beta_{8}) q^{70} +(\)\(10\!\cdots\!72\)\( + \)\(38\!\cdots\!08\)\( \beta_{1} + \)\(22\!\cdots\!72\)\( \beta_{2} - \)\(34\!\cdots\!96\)\( \beta_{3} - \)\(54\!\cdots\!31\)\( \beta_{4} + \)\(93\!\cdots\!12\)\( \beta_{5} - \)\(12\!\cdots\!85\)\( \beta_{6} + \)\(17\!\cdots\!48\)\( \beta_{7} + \)\(68\!\cdots\!24\)\( \beta_{8}) q^{71} +(\)\(29\!\cdots\!40\)\( + \)\(10\!\cdots\!21\)\( \beta_{1} + \)\(14\!\cdots\!40\)\( \beta_{2} + \)\(28\!\cdots\!01\)\( \beta_{3} + \)\(46\!\cdots\!10\)\( \beta_{4} + \)\(61\!\cdots\!47\)\( \beta_{5} + \)\(83\!\cdots\!88\)\( \beta_{6} - \)\(21\!\cdots\!35\)\( \beta_{7} + \)\(68\!\cdots\!00\)\( \beta_{8}) q^{72} +(\)\(75\!\cdots\!58\)\( + \)\(96\!\cdots\!44\)\( \beta_{1} + \)\(23\!\cdots\!60\)\( \beta_{2} + \)\(23\!\cdots\!36\)\( \beta_{3} + \)\(89\!\cdots\!36\)\( \beta_{4} + \)\(24\!\cdots\!83\)\( \beta_{5} - \)\(14\!\cdots\!93\)\( \beta_{6} - \)\(42\!\cdots\!14\)\( \beta_{7} - \)\(25\!\cdots\!75\)\( \beta_{8}) q^{73} +(\)\(24\!\cdots\!36\)\( + \)\(51\!\cdots\!70\)\( \beta_{1} - \)\(24\!\cdots\!48\)\( \beta_{2} + \)\(53\!\cdots\!48\)\( \beta_{3} + \)\(20\!\cdots\!24\)\( \beta_{4} - \)\(14\!\cdots\!00\)\( \beta_{5} + \)\(80\!\cdots\!20\)\( \beta_{6} + \)\(14\!\cdots\!84\)\( \beta_{7} - \)\(25\!\cdots\!08\)\( \beta_{8}) q^{74} +(\)\(20\!\cdots\!00\)\( - \)\(59\!\cdots\!45\)\( \beta_{1} - \)\(17\!\cdots\!65\)\( \beta_{2} - \)\(19\!\cdots\!20\)\( \beta_{3} - \)\(17\!\cdots\!80\)\( \beta_{4} + \)\(54\!\cdots\!00\)\( \beta_{5} + \)\(34\!\cdots\!60\)\( \beta_{6} - \)\(75\!\cdots\!00\)\( \beta_{7} + \)\(20\!\cdots\!20\)\( \beta_{8}) q^{75} +(\)\(84\!\cdots\!60\)\( - \)\(31\!\cdots\!80\)\( \beta_{1} - \)\(25\!\cdots\!08\)\( \beta_{2} + \)\(84\!\cdots\!12\)\( \beta_{3} - \)\(20\!\cdots\!04\)\( \beta_{4} + \)\(63\!\cdots\!76\)\( \beta_{5} - \)\(47\!\cdots\!60\)\( \beta_{6} - \)\(65\!\cdots\!52\)\( \beta_{7} - \)\(17\!\cdots\!76\)\( \beta_{8}) q^{76} +(\)\(15\!\cdots\!08\)\( - \)\(29\!\cdots\!96\)\( \beta_{1} + \)\(29\!\cdots\!40\)\( \beta_{2} - \)\(69\!\cdots\!68\)\( \beta_{3} + \)\(57\!\cdots\!00\)\( \beta_{4} - \)\(83\!\cdots\!66\)\( \beta_{5} + \)\(87\!\cdots\!86\)\( \beta_{6} + \)\(19\!\cdots\!60\)\( \beta_{7} - \)\(11\!\cdots\!50\)\( \beta_{8}) q^{77} +(\)\(11\!\cdots\!36\)\( - \)\(38\!\cdots\!22\)\( \beta_{1} + \)\(18\!\cdots\!16\)\( \beta_{2} + \)\(24\!\cdots\!02\)\( \beta_{3} - \)\(29\!\cdots\!48\)\( \beta_{4} - \)\(66\!\cdots\!94\)\( \beta_{5} - \)\(45\!\cdots\!76\)\( \beta_{6} - \)\(15\!\cdots\!48\)\( \beta_{7} + \)\(40\!\cdots\!00\)\( \beta_{8}) q^{78} +(\)\(83\!\cdots\!80\)\( - \)\(53\!\cdots\!56\)\( \beta_{1} + \)\(17\!\cdots\!04\)\( \beta_{2} - \)\(21\!\cdots\!84\)\( \beta_{3} + \)\(34\!\cdots\!14\)\( \beta_{4} - \)\(16\!\cdots\!76\)\( \beta_{5} + \)\(98\!\cdots\!10\)\( \beta_{6} - \)\(33\!\cdots\!88\)\( \beta_{7} - \)\(30\!\cdots\!44\)\( \beta_{8}) q^{79} +(\)\(36\!\cdots\!20\)\( + \)\(46\!\cdots\!64\)\( \beta_{1} - \)\(51\!\cdots\!08\)\( \beta_{2} + \)\(79\!\cdots\!72\)\( \beta_{3} - \)\(34\!\cdots\!56\)\( \beta_{4} + \)\(14\!\cdots\!00\)\( \beta_{5} + \)\(37\!\cdots\!60\)\( \beta_{6} + \)\(10\!\cdots\!00\)\( \beta_{7} - \)\(14\!\cdots\!80\)\( \beta_{8}) q^{80} +(\)\(54\!\cdots\!61\)\( + \)\(52\!\cdots\!40\)\( \beta_{1} - \)\(48\!\cdots\!60\)\( \beta_{2} - \)\(28\!\cdots\!72\)\( \beta_{3} + \)\(27\!\cdots\!04\)\( \beta_{4} - \)\(13\!\cdots\!43\)\( \beta_{5} - \)\(56\!\cdots\!35\)\( \beta_{6} - \)\(12\!\cdots\!02\)\( \beta_{7} + \)\(49\!\cdots\!99\)\( \beta_{8}) q^{81} +(\)\(12\!\cdots\!88\)\( + \)\(15\!\cdots\!50\)\( \beta_{1} - \)\(14\!\cdots\!24\)\( \beta_{2} + \)\(13\!\cdots\!20\)\( \beta_{3} + \)\(10\!\cdots\!88\)\( \beta_{4} - \)\(35\!\cdots\!52\)\( \beta_{5} + \)\(93\!\cdots\!92\)\( \beta_{6} + \)\(13\!\cdots\!48\)\( \beta_{7} - \)\(44\!\cdots\!00\)\( \beta_{8}) q^{82} +(\)\(22\!\cdots\!48\)\( - \)\(78\!\cdots\!43\)\( \beta_{1} - \)\(64\!\cdots\!47\)\( \beta_{2} + \)\(52\!\cdots\!60\)\( \beta_{3} - \)\(10\!\cdots\!60\)\( \beta_{4} + \)\(67\!\cdots\!60\)\( \beta_{5} - \)\(92\!\cdots\!60\)\( \beta_{6} - \)\(28\!\cdots\!60\)\( \beta_{7} - \)\(11\!\cdots\!00\)\( \beta_{8}) q^{83} +(\)\(97\!\cdots\!36\)\( - \)\(58\!\cdots\!84\)\( \beta_{1} + \)\(97\!\cdots\!20\)\( \beta_{2} + \)\(15\!\cdots\!24\)\( \beta_{3} - \)\(10\!\cdots\!24\)\( \beta_{4} - \)\(11\!\cdots\!88\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6} - \)\(28\!\cdots\!40\)\( \beta_{7} + \)\(41\!\cdots\!80\)\( \beta_{8}) q^{84} +(\)\(78\!\cdots\!20\)\( - \)\(79\!\cdots\!88\)\( \beta_{1} + \)\(52\!\cdots\!70\)\( \beta_{2} - \)\(14\!\cdots\!96\)\( \beta_{3} - \)\(69\!\cdots\!50\)\( \beta_{4} + \)\(18\!\cdots\!50\)\( \beta_{5} + \)\(99\!\cdots\!82\)\( \beta_{6} + \)\(40\!\cdots\!80\)\( \beta_{7} - \)\(41\!\cdots\!06\)\( \beta_{8}) q^{85} +(\)\(83\!\cdots\!92\)\( - \)\(38\!\cdots\!49\)\( \beta_{1} + \)\(18\!\cdots\!68\)\( \beta_{2} - \)\(27\!\cdots\!29\)\( \beta_{3} + \)\(19\!\cdots\!82\)\( \beta_{4} - \)\(77\!\cdots\!03\)\( \beta_{5} - \)\(24\!\cdots\!20\)\( \beta_{6} - \)\(96\!\cdots\!24\)\( \beta_{7} - \)\(58\!\cdots\!12\)\( \beta_{8}) q^{86} +(\)\(31\!\cdots\!40\)\( - \)\(10\!\cdots\!92\)\( \beta_{1} - \)\(50\!\cdots\!24\)\( \beta_{2} - \)\(83\!\cdots\!08\)\( \beta_{3} + \)\(70\!\cdots\!31\)\( \beta_{4} + \)\(28\!\cdots\!00\)\( \beta_{5} + \)\(19\!\cdots\!25\)\( \beta_{6} + \)\(10\!\cdots\!36\)\( \beta_{7} + \)\(23\!\cdots\!00\)\( \beta_{8}) q^{87} +(\)\(79\!\cdots\!40\)\( + \)\(91\!\cdots\!32\)\( \beta_{1} - \)\(14\!\cdots\!36\)\( \beta_{2} + \)\(26\!\cdots\!92\)\( \beta_{3} + \)\(11\!\cdots\!80\)\( \beta_{4} + \)\(24\!\cdots\!84\)\( \beta_{5} - \)\(25\!\cdots\!64\)\( \beta_{6} + \)\(40\!\cdots\!40\)\( \beta_{7} - \)\(22\!\cdots\!00\)\( \beta_{8}) q^{88} +(\)\(41\!\cdots\!90\)\( + \)\(19\!\cdots\!72\)\( \beta_{1} + \)\(26\!\cdots\!36\)\( \beta_{2} + \)\(27\!\cdots\!04\)\( \beta_{3} - \)\(12\!\cdots\!80\)\( \beta_{4} - \)\(30\!\cdots\!33\)\( \beta_{5} - \)\(39\!\cdots\!05\)\( \beta_{6} - \)\(75\!\cdots\!26\)\( \beta_{7} - \)\(15\!\cdots\!63\)\( \beta_{8}) q^{89} +(\)\(99\!\cdots\!40\)\( + \)\(23\!\cdots\!74\)\( \beta_{1} + \)\(22\!\cdots\!40\)\( \beta_{2} + \)\(35\!\cdots\!08\)\( \beta_{3} + \)\(51\!\cdots\!00\)\( \beta_{4} - \)\(83\!\cdots\!00\)\( \beta_{5} + \)\(98\!\cdots\!64\)\( \beta_{6} - \)\(21\!\cdots\!40\)\( \beta_{7} + \)\(56\!\cdots\!88\)\( \beta_{8}) q^{90} +(-\)\(28\!\cdots\!28\)\( + \)\(57\!\cdots\!40\)\( \beta_{1} + \)\(65\!\cdots\!76\)\( \beta_{2} - \)\(92\!\cdots\!64\)\( \beta_{3} + \)\(48\!\cdots\!08\)\( \beta_{4} - \)\(58\!\cdots\!92\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6} + \)\(26\!\cdots\!24\)\( \beta_{7} - \)\(33\!\cdots\!88\)\( \beta_{8}) q^{91} +(-\)\(18\!\cdots\!76\)\( - \)\(53\!\cdots\!36\)\( \beta_{1} + \)\(31\!\cdots\!64\)\( \beta_{2} - \)\(55\!\cdots\!40\)\( \beta_{3} - \)\(40\!\cdots\!12\)\( \beta_{4} + \)\(24\!\cdots\!28\)\( \beta_{5} - \)\(35\!\cdots\!88\)\( \beta_{6} - \)\(24\!\cdots\!52\)\( \beta_{7} + \)\(47\!\cdots\!00\)\( \beta_{8}) q^{92} +(-\)\(61\!\cdots\!04\)\( - \)\(16\!\cdots\!60\)\( \beta_{1} - \)\(22\!\cdots\!80\)\( \beta_{2} + \)\(19\!\cdots\!08\)\( \beta_{3} - \)\(32\!\cdots\!32\)\( \beta_{4} - \)\(20\!\cdots\!16\)\( \beta_{5} - \)\(61\!\cdots\!64\)\( \beta_{6} - \)\(50\!\cdots\!32\)\( \beta_{7} - \)\(36\!\cdots\!00\)\( \beta_{8}) q^{93} +(-\)\(79\!\cdots\!04\)\( + \)\(75\!\cdots\!88\)\( \beta_{1} - \)\(55\!\cdots\!80\)\( \beta_{2} + \)\(16\!\cdots\!24\)\( \beta_{3} - \)\(15\!\cdots\!16\)\( \beta_{4} - \)\(57\!\cdots\!96\)\( \beta_{5} + \)\(26\!\cdots\!60\)\( \beta_{6} + \)\(11\!\cdots\!92\)\( \beta_{7} + \)\(62\!\cdots\!96\)\( \beta_{8}) q^{94} +(-\)\(18\!\cdots\!00\)\( - \)\(32\!\cdots\!00\)\( \beta_{1} - \)\(20\!\cdots\!40\)\( \beta_{2} + \)\(76\!\cdots\!20\)\( \beta_{3} + \)\(37\!\cdots\!95\)\( \beta_{4} + \)\(22\!\cdots\!00\)\( \beta_{5} - \)\(14\!\cdots\!95\)\( \beta_{6} + \)\(36\!\cdots\!00\)\( \beta_{7} + \)\(89\!\cdots\!60\)\( \beta_{8}) q^{95} +(-\)\(13\!\cdots\!08\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!36\)\( \beta_{2} - \)\(23\!\cdots\!88\)\( \beta_{3} - \)\(24\!\cdots\!04\)\( \beta_{4} - \)\(44\!\cdots\!48\)\( \beta_{5} - \)\(54\!\cdots\!00\)\( \beta_{6} - \)\(38\!\cdots\!40\)\( \beta_{7} - \)\(49\!\cdots\!20\)\( \beta_{8}) q^{96} +(-\)\(70\!\cdots\!06\)\( + \)\(93\!\cdots\!20\)\( \beta_{1} + \)\(17\!\cdots\!32\)\( \beta_{2} + \)\(75\!\cdots\!40\)\( \beta_{3} + \)\(31\!\cdots\!84\)\( \beta_{4} + \)\(55\!\cdots\!99\)\( \beta_{5} + \)\(98\!\cdots\!71\)\( \beta_{6} + \)\(35\!\cdots\!14\)\( \beta_{7} + \)\(58\!\cdots\!25\)\( \beta_{8}) q^{97} +(-\)\(21\!\cdots\!48\)\( + \)\(31\!\cdots\!41\)\( \beta_{1} + \)\(28\!\cdots\!96\)\( \beta_{2} - \)\(46\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!52\)\( \beta_{4} - \)\(75\!\cdots\!32\)\( \beta_{5} - \)\(78\!\cdots\!28\)\( \beta_{6} + \)\(41\!\cdots\!08\)\( \beta_{7} + \)\(62\!\cdots\!00\)\( \beta_{8}) q^{98} +(-\)\(33\!\cdots\!16\)\( + \)\(39\!\cdots\!95\)\( \beta_{1} + \)\(39\!\cdots\!59\)\( \beta_{2} + \)\(12\!\cdots\!64\)\( \beta_{3} - \)\(28\!\cdots\!68\)\( \beta_{4} - \)\(78\!\cdots\!08\)\( \beta_{5} + \)\(16\!\cdots\!80\)\( \beta_{6} - \)\(10\!\cdots\!84\)\( \beta_{7} - \)\(25\!\cdots\!92\)\( \beta_{8}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 7300500744476376q^{2} + \)\(23\!\cdots\!52\)\(q^{3} + \)\(11\!\cdots\!52\)\(q^{4} + \)\(81\!\cdots\!30\)\(q^{5} - \)\(21\!\cdots\!32\)\(q^{6} + \)\(78\!\cdots\!56\)\(q^{7} + \)\(33\!\cdots\!80\)\(q^{8} + \)\(44\!\cdots\!13\)\(q^{9} + O(q^{10}) \) \( 9q + 7300500744476376q^{2} + \)\(23\!\cdots\!52\)\(q^{3} + \)\(11\!\cdots\!52\)\(q^{4} + \)\(81\!\cdots\!30\)\(q^{5} - \)\(21\!\cdots\!32\)\(q^{6} + \)\(78\!\cdots\!56\)\(q^{7} + \)\(33\!\cdots\!80\)\(q^{8} + \)\(44\!\cdots\!13\)\(q^{9} + \)\(70\!\cdots\!80\)\(q^{10} + \)\(94\!\cdots\!08\)\(q^{11} + \)\(62\!\cdots\!56\)\(q^{12} - \)\(10\!\cdots\!38\)\(q^{13} - \)\(69\!\cdots\!96\)\(q^{14} - \)\(26\!\cdots\!40\)\(q^{15} + \)\(27\!\cdots\!24\)\(q^{16} - \)\(22\!\cdots\!34\)\(q^{17} + \)\(13\!\cdots\!32\)\(q^{18} - \)\(84\!\cdots\!20\)\(q^{19} + \)\(77\!\cdots\!40\)\(q^{20} + \)\(53\!\cdots\!08\)\(q^{21} + \)\(13\!\cdots\!12\)\(q^{22} - \)\(42\!\cdots\!28\)\(q^{23} - \)\(12\!\cdots\!60\)\(q^{24} - \)\(45\!\cdots\!25\)\(q^{25} + \)\(91\!\cdots\!08\)\(q^{26} + \)\(36\!\cdots\!20\)\(q^{27} + \)\(38\!\cdots\!68\)\(q^{28} - \)\(16\!\cdots\!30\)\(q^{29} - \)\(18\!\cdots\!40\)\(q^{30} + \)\(67\!\cdots\!88\)\(q^{31} + \)\(75\!\cdots\!96\)\(q^{32} - \)\(31\!\cdots\!76\)\(q^{33} - \)\(24\!\cdots\!56\)\(q^{34} + \)\(57\!\cdots\!80\)\(q^{35} + \)\(32\!\cdots\!64\)\(q^{36} + \)\(30\!\cdots\!86\)\(q^{37} - \)\(21\!\cdots\!80\)\(q^{38} + \)\(32\!\cdots\!16\)\(q^{39} + \)\(12\!\cdots\!00\)\(q^{40} + \)\(10\!\cdots\!78\)\(q^{41} + \)\(20\!\cdots\!12\)\(q^{42} - \)\(19\!\cdots\!08\)\(q^{43} + \)\(54\!\cdots\!24\)\(q^{44} + \)\(15\!\cdots\!10\)\(q^{45} - \)\(39\!\cdots\!52\)\(q^{46} + \)\(14\!\cdots\!96\)\(q^{47} - \)\(74\!\cdots\!28\)\(q^{48} + \)\(17\!\cdots\!37\)\(q^{49} + \)\(20\!\cdots\!00\)\(q^{50} - \)\(85\!\cdots\!12\)\(q^{51} - \)\(22\!\cdots\!64\)\(q^{52} - \)\(33\!\cdots\!98\)\(q^{53} + \)\(45\!\cdots\!80\)\(q^{54} + \)\(54\!\cdots\!60\)\(q^{55} - \)\(42\!\cdots\!80\)\(q^{56} + \)\(99\!\cdots\!40\)\(q^{57} + \)\(32\!\cdots\!80\)\(q^{58} + \)\(32\!\cdots\!40\)\(q^{59} - \)\(13\!\cdots\!20\)\(q^{60} - \)\(17\!\cdots\!42\)\(q^{61} - \)\(17\!\cdots\!68\)\(q^{62} + \)\(12\!\cdots\!92\)\(q^{63} + \)\(14\!\cdots\!12\)\(q^{64} - \)\(18\!\cdots\!40\)\(q^{65} - \)\(10\!\cdots\!84\)\(q^{66} + \)\(13\!\cdots\!16\)\(q^{67} + \)\(11\!\cdots\!48\)\(q^{68} + \)\(10\!\cdots\!96\)\(q^{69} - \)\(36\!\cdots\!20\)\(q^{70} + \)\(98\!\cdots\!48\)\(q^{71} + \)\(26\!\cdots\!60\)\(q^{72} + \)\(68\!\cdots\!22\)\(q^{73} + \)\(22\!\cdots\!24\)\(q^{74} + \)\(18\!\cdots\!00\)\(q^{75} + \)\(76\!\cdots\!40\)\(q^{76} + \)\(13\!\cdots\!72\)\(q^{77} + \)\(10\!\cdots\!24\)\(q^{78} + \)\(75\!\cdots\!20\)\(q^{79} + \)\(32\!\cdots\!80\)\(q^{80} + \)\(48\!\cdots\!49\)\(q^{81} + \)\(11\!\cdots\!92\)\(q^{82} + \)\(19\!\cdots\!32\)\(q^{83} + \)\(88\!\cdots\!24\)\(q^{84} + \)\(70\!\cdots\!80\)\(q^{85} + \)\(74\!\cdots\!28\)\(q^{86} + \)\(28\!\cdots\!60\)\(q^{87} + \)\(71\!\cdots\!60\)\(q^{88} + \)\(37\!\cdots\!10\)\(q^{89} + \)\(89\!\cdots\!60\)\(q^{90} - \)\(25\!\cdots\!52\)\(q^{91} - \)\(16\!\cdots\!84\)\(q^{92} - \)\(55\!\cdots\!36\)\(q^{93} - \)\(71\!\cdots\!36\)\(q^{94} - \)\(16\!\cdots\!00\)\(q^{95} - \)\(11\!\cdots\!72\)\(q^{96} - \)\(63\!\cdots\!54\)\(q^{97} - \)\(18\!\cdots\!32\)\(q^{98} - \)\(30\!\cdots\!44\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - \)\(33\!\cdots\!40\)\( x^{7} - \)\(25\!\cdots\!60\)\( x^{6} + \)\(33\!\cdots\!30\)\( x^{5} + \)\(48\!\cdots\!24\)\( x^{4} - \)\(99\!\cdots\!80\)\( x^{3} - \)\(21\!\cdots\!00\)\( x^{2} + \)\(25\!\cdots\!25\)\( x + \)\(83\!\cdots\!00\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(22\!\cdots\!23\)\( \nu^{8} + \)\(13\!\cdots\!91\)\( \nu^{7} - \)\(77\!\cdots\!05\)\( \nu^{6} - \)\(53\!\cdots\!25\)\( \nu^{5} + \)\(84\!\cdots\!93\)\( \nu^{4} + \)\(61\!\cdots\!73\)\( \nu^{3} - \)\(27\!\cdots\!91\)\( \nu^{2} - \)\(18\!\cdots\!99\)\( \nu - \)\(73\!\cdots\!80\)\(\)\()/ \)\(38\!\cdots\!52\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(24\!\cdots\!99\)\( \nu^{8} - \)\(14\!\cdots\!83\)\( \nu^{7} + \)\(85\!\cdots\!65\)\( \nu^{6} + \)\(58\!\cdots\!25\)\( \nu^{5} - \)\(92\!\cdots\!09\)\( \nu^{4} - \)\(67\!\cdots\!49\)\( \nu^{3} + \)\(79\!\cdots\!75\)\( \nu^{2} + \)\(14\!\cdots\!07\)\( \nu - \)\(36\!\cdots\!00\)\(\)\()/ \)\(95\!\cdots\!88\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(39\!\cdots\!89\)\( \nu^{8} + \)\(27\!\cdots\!31\)\( \nu^{7} + \)\(13\!\cdots\!11\)\( \nu^{6} + \)\(13\!\cdots\!71\)\( \nu^{5} - \)\(13\!\cdots\!79\)\( \nu^{4} - \)\(10\!\cdots\!95\)\( \nu^{3} + \)\(43\!\cdots\!25\)\( \nu^{2} + \)\(61\!\cdots\!25\)\( \nu - \)\(12\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(44\!\cdots\!81\)\( \nu^{8} + \)\(24\!\cdots\!99\)\( \nu^{7} + \)\(15\!\cdots\!19\)\( \nu^{6} - \)\(70\!\cdots\!41\)\( \nu^{5} - \)\(16\!\cdots\!91\)\( \nu^{4} + \)\(52\!\cdots\!45\)\( \nu^{3} + \)\(43\!\cdots\!25\)\( \nu^{2} - \)\(68\!\cdots\!75\)\( \nu + \)\(43\!\cdots\!00\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(10\!\cdots\!03\)\( \nu^{8} + \)\(22\!\cdots\!37\)\( \nu^{7} + \)\(24\!\cdots\!97\)\( \nu^{6} - \)\(62\!\cdots\!83\)\( \nu^{5} - \)\(12\!\cdots\!33\)\( \nu^{4} + \)\(49\!\cdots\!35\)\( \nu^{3} - \)\(14\!\cdots\!25\)\( \nu^{2} - \)\(10\!\cdots\!25\)\( \nu + \)\(10\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(12\!\cdots\!89\)\( \nu^{8} - \)\(49\!\cdots\!31\)\( \nu^{7} - \)\(41\!\cdots\!11\)\( \nu^{6} + \)\(10\!\cdots\!29\)\( \nu^{5} + \)\(40\!\cdots\!79\)\( \nu^{4} - \)\(27\!\cdots\!05\)\( \nu^{3} - \)\(11\!\cdots\!25\)\( \nu^{2} - \)\(30\!\cdots\!25\)\( \nu + \)\(88\!\cdots\!00\)\(\)\()/ \)\(52\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(22\!\cdots\!33\)\( \nu^{8} + \)\(15\!\cdots\!07\)\( \nu^{7} + \)\(71\!\cdots\!87\)\( \nu^{6} + \)\(67\!\cdots\!87\)\( \nu^{5} - \)\(68\!\cdots\!23\)\( \nu^{4} - \)\(59\!\cdots\!35\)\( \nu^{3} + \)\(18\!\cdots\!65\)\( \nu^{2} + \)\(34\!\cdots\!65\)\( \nu - \)\(16\!\cdots\!00\)\(\)\()/ \)\(19\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 439252 \beta_{2} + 8323470902756930 \beta_{1} + 3819942678867500861095870306183680\)\()/5184\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 544 \beta_{6} + 1158411 \beta_{5} + 36618818906 \beta_{4} + 4992822124202353 \beta_{3} - 83151424381156719106124 \beta_{2} + 6808358022966235381099892796225629 \beta_{1} + 31795181737658367388374525999206626071736123944960\)\()/373248\)
\(\nu^{4}\)\(=\)\((\)\(2151791746368 \beta_{8} + 2152381264793679 \beta_{7} + 70870079271834272 \beta_{6} + 36429483990796555689125 \beta_{5} - 986286471004806407707271610 \beta_{4} + 1113684925644086232461500883274951 \beta_{3} - 256803729326628582757925257100468353492 \beta_{2} + 12709404098708678753562414927036412121276154827075 \beta_{1} + 3250942173119574726797614605721862891093054017685950667143797411840\)\()/3359232\)
\(\nu^{5}\)\(=\)\((\)\(-1764772133542641013345349760 \beta_{8} + 85831786946306258073818340395239 \beta_{7} + 59832405271817422263091935415405408 \beta_{6} + 127024762499748007368624918930599510637 \beta_{5} + 2027232966422706889944580372999418068599094 \beta_{4} + 941587397453036682542535295576027025159416732599 \beta_{3} - 13790715262261783933734370436650971589517376939767346324 \beta_{2} + 412237837267741061446098679275699317383068447839293998337259248075 \beta_{1} + 3034324696230384552661843694238344134793168541411825619078448223996186610876078080\)\()/15116544\)
\(\nu^{6}\)\(=\)\((\)\(\)\(17\!\cdots\!20\)\( \beta_{8} + \)\(30\!\cdots\!77\)\( \beta_{7} + \)\(24\!\cdots\!64\)\( \beta_{6} + \)\(48\!\cdots\!03\)\( \beta_{5} - \)\(92\!\cdots\!82\)\( \beta_{4} + \)\(83\!\cdots\!33\)\( \beta_{3} - \)\(65\!\cdots\!48\)\( \beta_{2} + \)\(12\!\cdots\!93\)\( \beta_{1} + \)\(21\!\cdots\!80\)\(\)\()/15116544\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(14\!\cdots\!40\)\( \beta_{8} + \)\(31\!\cdots\!51\)\( \beta_{7} + \)\(26\!\cdots\!00\)\( \beta_{6} + \)\(61\!\cdots\!89\)\( \beta_{5} + \)\(50\!\cdots\!98\)\( \beta_{4} + \)\(51\!\cdots\!75\)\( \beta_{3} - \)\(66\!\cdots\!20\)\( \beta_{2} + \)\(13\!\cdots\!59\)\( \beta_{1} + \)\(13\!\cdots\!60\)\(\)\()/30233088\)
\(\nu^{8}\)\(=\)\((\)\(\)\(16\!\cdots\!96\)\( \beta_{8} + \)\(46\!\cdots\!47\)\( \beta_{7} + \)\(79\!\cdots\!16\)\( \beta_{6} + \)\(71\!\cdots\!77\)\( \beta_{5} - \)\(10\!\cdots\!34\)\( \beta_{4} + \)\(94\!\cdots\!39\)\( \beta_{3} - \)\(12\!\cdots\!76\)\( \beta_{2} + \)\(17\!\cdots\!35\)\( \beta_{1} + \)\(23\!\cdots\!20\)\(\)\()/10077696\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.25626e15
−1.03033e15
−5.88839e14
−3.56630e14
−2.69685e13
1.08230e14
7.31006e14
1.06213e15
1.35766e15
−8.96393e16 3.53927e26 5.43906e33 4.17124e38 −3.17258e43 1.21017e47 −2.54837e50 3.39667e52 −3.73907e55
1.2 −7.33726e16 −2.81001e26 2.78739e33 1.03167e38 2.06178e43 −1.16535e47 −1.40317e49 −1.23360e52 −7.56964e54
1.3 −4.15853e16 −6.11584e25 −8.66814e32 −6.51103e38 2.54329e42 1.10096e47 1.44008e50 −8.75572e52 2.70763e55
1.4 −2.48662e16 5.20768e26 −1.97782e33 −7.34308e38 −1.29495e43 −1.22695e47 1.13737e50 1.79902e53 1.82594e55
1.5 −1.13057e15 1.34574e26 −2.59487e33 1.00336e39 −1.52145e41 −8.67417e45 5.86879e48 −7.31873e52 −1.13437e54
1.6 8.60376e15 −5.90178e26 −2.52212e33 1.16842e38 −5.07775e42 5.43178e46 −4.40364e49 2.57013e53 1.00528e54
1.7 5.34436e16 −3.57964e25 2.60068e32 −4.56192e38 −1.91309e42 −2.39634e46 −1.24849e50 −9.00162e52 −2.43805e55
1.8 7.72845e16 5.46172e26 3.37675e33 2.64030e38 4.22106e43 1.15348e47 6.03282e49 2.07006e53 2.04054e55
1.9 9.85626e16 −3.53632e26 7.11844e33 7.51271e38 −3.48549e43 −5.05213e46 4.45728e50 3.37583e52 7.40472e55
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Hecke kernels

There are no other newforms in \(S_{112}^{\mathrm{new}}(\Gamma_0(1))\).