Properties

Label 1.120.a.a
Level 1
Weight 120
Character orbit 1.a
Self dual Yes
Analytic conductor 89.678
Analytic rank 0
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 120 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(89.677690876\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{171}\cdot 3^{61}\cdot 5^{22}\cdot 7^{9}\cdot 11^{6}\cdot 13^{3}\cdot 17^{4} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(91993272260576940 - \beta_{1}) q^{2} +(-\)\(95\!\cdots\!60\)\( - 3441124586 \beta_{1} + \beta_{2}) q^{3} +(\)\(42\!\cdots\!28\)\( + 32048935683288551 \beta_{1} - 4314966 \beta_{2} + \beta_{3}) q^{4} +(\)\(60\!\cdots\!34\)\( - \)\(53\!\cdots\!95\)\( \beta_{1} - 666477877108 \beta_{2} + 68414 \beta_{3} - \beta_{4}) q^{5} +(\)\(36\!\cdots\!12\)\( + \)\(44\!\cdots\!02\)\( \beta_{1} + 141054558156867115 \beta_{2} + 18058531172 \beta_{3} - 7345 \beta_{4} + \beta_{5}) q^{6} +(-\)\(21\!\cdots\!00\)\( - \)\(13\!\cdots\!47\)\( \beta_{1} + \)\(29\!\cdots\!51\)\( \beta_{2} - 59589858539313 \beta_{3} - 94562555 \beta_{4} - 1153 \beta_{5} - \beta_{6}) q^{7} +(-\)\(56\!\cdots\!20\)\( - \)\(42\!\cdots\!96\)\( \beta_{1} + \)\(12\!\cdots\!14\)\( \beta_{2} + 930910859864949 \beta_{3} - 285635101119 \beta_{4} - 4418884 \beta_{5} + 194 \beta_{6} - \beta_{7}) q^{8} +(\)\(18\!\cdots\!37\)\( - \)\(15\!\cdots\!50\)\( \beta_{1} - \)\(65\!\cdots\!94\)\( \beta_{2} + \)\(13\!\cdots\!71\)\( \beta_{3} - 416842889875292 \beta_{4} - 325296740 \beta_{5} - 17503 \beta_{6} + 211 \beta_{7} + \beta_{8}) q^{9} +O(q^{10})\) \( q +(91993272260576940 - \beta_{1}) q^{2} +(-\)\(95\!\cdots\!60\)\( - 3441124586 \beta_{1} + \beta_{2}) q^{3} +(\)\(42\!\cdots\!28\)\( + 32048935683288551 \beta_{1} - 4314966 \beta_{2} + \beta_{3}) q^{4} +(\)\(60\!\cdots\!34\)\( - \)\(53\!\cdots\!95\)\( \beta_{1} - 666477877108 \beta_{2} + 68414 \beta_{3} - \beta_{4}) q^{5} +(\)\(36\!\cdots\!12\)\( + \)\(44\!\cdots\!02\)\( \beta_{1} + 141054558156867115 \beta_{2} + 18058531172 \beta_{3} - 7345 \beta_{4} + \beta_{5}) q^{6} +(-\)\(21\!\cdots\!00\)\( - \)\(13\!\cdots\!47\)\( \beta_{1} + \)\(29\!\cdots\!51\)\( \beta_{2} - 59589858539313 \beta_{3} - 94562555 \beta_{4} - 1153 \beta_{5} - \beta_{6}) q^{7} +(-\)\(56\!\cdots\!20\)\( - \)\(42\!\cdots\!96\)\( \beta_{1} + \)\(12\!\cdots\!14\)\( \beta_{2} + 930910859864949 \beta_{3} - 285635101119 \beta_{4} - 4418884 \beta_{5} + 194 \beta_{6} - \beta_{7}) q^{8} +(\)\(18\!\cdots\!37\)\( - \)\(15\!\cdots\!50\)\( \beta_{1} - \)\(65\!\cdots\!94\)\( \beta_{2} + \)\(13\!\cdots\!71\)\( \beta_{3} - 416842889875292 \beta_{4} - 325296740 \beta_{5} - 17503 \beta_{6} + 211 \beta_{7} + \beta_{8}) q^{9} +(\)\(63\!\cdots\!04\)\( - \)\(10\!\cdots\!64\)\( \beta_{1} + \)\(23\!\cdots\!03\)\( \beta_{2} + \)\(13\!\cdots\!70\)\( \beta_{3} - 364646369225594070 \beta_{4} - 402339533945 \beta_{5} - 337766567 \beta_{6} - 102886 \beta_{7} + 394 \beta_{8} - \beta_{9}) q^{10} +(\)\(10\!\cdots\!32\)\( - \)\(15\!\cdots\!40\)\( \beta_{1} - \)\(52\!\cdots\!59\)\( \beta_{2} - \)\(97\!\cdots\!66\)\( \beta_{3} - 9259802188481904958 \beta_{4} - 365106088800674 \beta_{5} + 76668709018 \beta_{6} - 27766556 \beta_{7} + 73164 \beta_{8} + 240 \beta_{9}) q^{11} +(-\)\(38\!\cdots\!20\)\( - \)\(13\!\cdots\!96\)\( \beta_{1} + \)\(52\!\cdots\!80\)\( \beta_{2} - \)\(13\!\cdots\!56\)\( \beta_{3} - \)\(10\!\cdots\!64\)\( \beta_{4} - 128650493421700344 \beta_{5} - 23151651447816 \beta_{6} - 31806791136 \beta_{7} - 9945360 \beta_{8} - 28440 \beta_{9}) q^{12} +(-\)\(11\!\cdots\!70\)\( + \)\(19\!\cdots\!09\)\( \beta_{1} - \)\(48\!\cdots\!36\)\( \beta_{2} - \)\(47\!\cdots\!96\)\( \beta_{3} - \)\(30\!\cdots\!01\)\( \beta_{4} - 3545103753734169528 \beta_{5} - 5790885415566498 \beta_{6} + 136266654266 \beta_{7} + 27030430 \beta_{8} + 2217920 \beta_{9}) q^{13} +(\)\(12\!\cdots\!76\)\( + \)\(62\!\cdots\!00\)\( \beta_{1} - \)\(16\!\cdots\!10\)\( \beta_{2} + \)\(11\!\cdots\!68\)\( \beta_{3} - \)\(75\!\cdots\!06\)\( \beta_{4} + \)\(17\!\cdots\!66\)\( \beta_{5} - 727935258896143844 \beta_{6} + 171636923450008 \beta_{7} + 62753160408 \beta_{8} - 128013180 \beta_{9}) q^{14} +(-\)\(37\!\cdots\!32\)\( - \)\(19\!\cdots\!93\)\( \beta_{1} + \)\(16\!\cdots\!21\)\( \beta_{2} + \)\(42\!\cdots\!85\)\( \beta_{3} - \)\(10\!\cdots\!45\)\( \beta_{4} + \)\(25\!\cdots\!85\)\( \beta_{5} + 9539869587499913121 \beta_{6} - 9424204120636632 \beta_{7} - 5891215161672 \beta_{8} + 5830632288 \beta_{9}) q^{15} +(\)\(16\!\cdots\!56\)\( + \)\(12\!\cdots\!52\)\( \beta_{1} - \)\(40\!\cdots\!60\)\( \beta_{2} + \)\(49\!\cdots\!76\)\( \beta_{3} - \)\(24\!\cdots\!88\)\( \beta_{4} + \)\(20\!\cdots\!64\)\( \beta_{5} + \)\(12\!\cdots\!68\)\( \beta_{6} + 152195927046746824 \beta_{7} + 318100461732224 \beta_{8} - 218210671040 \beta_{9}) q^{16} +(-\)\(78\!\cdots\!30\)\( + \)\(37\!\cdots\!14\)\( \beta_{1} + \)\(12\!\cdots\!38\)\( \beta_{2} + \)\(19\!\cdots\!83\)\( \beta_{3} - \)\(85\!\cdots\!20\)\( \beta_{4} + \)\(29\!\cdots\!28\)\( \beta_{5} - \)\(32\!\cdots\!99\)\( \beta_{6} + 4870191598379980935 \beta_{7} - 12307319357674755 \beta_{8} + 6898816087680 \beta_{9}) q^{17} +(\)\(33\!\cdots\!40\)\( - \)\(29\!\cdots\!53\)\( \beta_{1} + \)\(12\!\cdots\!26\)\( \beta_{2} - \)\(14\!\cdots\!88\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4} - \)\(15\!\cdots\!18\)\( \beta_{5} - \)\(50\!\cdots\!86\)\( \beta_{6} - \)\(34\!\cdots\!80\)\( \beta_{7} + 370524547931222460 \beta_{8} - 187998704387910 \beta_{9}) q^{18} +(\)\(69\!\cdots\!80\)\( + \)\(32\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!21\)\( \beta_{2} - \)\(16\!\cdots\!34\)\( \beta_{3} - \)\(13\!\cdots\!34\)\( \beta_{4} - \)\(15\!\cdots\!62\)\( \beta_{5} + \)\(33\!\cdots\!14\)\( \beta_{6} + \)\(99\!\cdots\!12\)\( \beta_{7} - 9061587380913137628 \beta_{8} + 4483600695226320 \beta_{9}) q^{19} +(\)\(73\!\cdots\!32\)\( - \)\(94\!\cdots\!70\)\( \beta_{1} + \)\(24\!\cdots\!56\)\( \beta_{2} + \)\(18\!\cdots\!62\)\( \beta_{3} - \)\(67\!\cdots\!08\)\( \beta_{4} + \)\(50\!\cdots\!00\)\( \beta_{5} - \)\(53\!\cdots\!80\)\( \beta_{6} - \)\(17\!\cdots\!40\)\( \beta_{7} + \)\(18\!\cdots\!60\)\( \beta_{8} - 94696940080115040 \beta_{9}) q^{20} +(\)\(29\!\cdots\!52\)\( + \)\(14\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2} + \)\(17\!\cdots\!98\)\( \beta_{3} + \)\(32\!\cdots\!48\)\( \beta_{4} + \)\(57\!\cdots\!32\)\( \beta_{5} - \)\(31\!\cdots\!98\)\( \beta_{6} + \)\(17\!\cdots\!86\)\( \beta_{7} - \)\(31\!\cdots\!14\)\( \beta_{8} + 1788023593898504640 \beta_{9}) q^{21} +(\)\(18\!\cdots\!80\)\( - \)\(35\!\cdots\!46\)\( \beta_{1} - \)\(58\!\cdots\!91\)\( \beta_{2} + \)\(23\!\cdots\!64\)\( \beta_{3} + \)\(12\!\cdots\!09\)\( \beta_{4} - \)\(19\!\cdots\!13\)\( \beta_{5} + \)\(14\!\cdots\!52\)\( \beta_{6} + \)\(11\!\cdots\!76\)\( \beta_{7} + \)\(46\!\cdots\!20\)\( \beta_{8} - 30411800092600150520 \beta_{9}) q^{22} +(\)\(28\!\cdots\!20\)\( - \)\(44\!\cdots\!93\)\( \beta_{1} + \)\(35\!\cdots\!49\)\( \beta_{2} + \)\(93\!\cdots\!25\)\( \beta_{3} + \)\(60\!\cdots\!95\)\( \beta_{4} + \)\(16\!\cdots\!85\)\( \beta_{5} - \)\(23\!\cdots\!15\)\( \beta_{6} - \)\(41\!\cdots\!80\)\( \beta_{7} - \)\(59\!\cdots\!20\)\( \beta_{8} + \)\(46\!\cdots\!20\)\( \beta_{9}) q^{23} +(\)\(11\!\cdots\!00\)\( + \)\(41\!\cdots\!00\)\( \beta_{1} - \)\(16\!\cdots\!52\)\( \beta_{2} + \)\(26\!\cdots\!04\)\( \beta_{3} - \)\(19\!\cdots\!68\)\( \beta_{4} + \)\(34\!\cdots\!92\)\( \beta_{5} + \)\(13\!\cdots\!48\)\( \beta_{6} + \)\(89\!\cdots\!24\)\( \beta_{7} + \)\(63\!\cdots\!84\)\( \beta_{8} - \)\(65\!\cdots\!00\)\( \beta_{9}) q^{24} +(\)\(43\!\cdots\!35\)\( - \)\(59\!\cdots\!80\)\( \beta_{1} + \)\(47\!\cdots\!00\)\( \beta_{2} + \)\(88\!\cdots\!30\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4} - \)\(34\!\cdots\!00\)\( \beta_{5} + \)\(14\!\cdots\!10\)\( \beta_{6} - \)\(11\!\cdots\!70\)\( \beta_{7} - \)\(55\!\cdots\!70\)\( \beta_{8} + \)\(84\!\cdots\!80\)\( \beta_{9}) q^{25} +(-\)\(31\!\cdots\!28\)\( + \)\(40\!\cdots\!08\)\( \beta_{1} - \)\(96\!\cdots\!37\)\( \beta_{2} - \)\(23\!\cdots\!94\)\( \beta_{3} + \)\(14\!\cdots\!34\)\( \beta_{4} - \)\(83\!\cdots\!01\)\( \beta_{5} - \)\(38\!\cdots\!99\)\( \beta_{6} + \)\(11\!\cdots\!38\)\( \beta_{7} + \)\(38\!\cdots\!58\)\( \beta_{8} - \)\(99\!\cdots\!25\)\( \beta_{9}) q^{26} +(-\)\(45\!\cdots\!80\)\( - \)\(20\!\cdots\!74\)\( \beta_{1} + \)\(22\!\cdots\!16\)\( \beta_{2} - \)\(54\!\cdots\!34\)\( \beta_{3} + \)\(14\!\cdots\!34\)\( \beta_{4} + \)\(35\!\cdots\!34\)\( \beta_{5} + \)\(37\!\cdots\!86\)\( \beta_{6} - \)\(63\!\cdots\!04\)\( \beta_{7} - \)\(15\!\cdots\!80\)\( \beta_{8} + \)\(10\!\cdots\!80\)\( \beta_{9}) q^{27} +(-\)\(52\!\cdots\!20\)\( - \)\(24\!\cdots\!28\)\( \beta_{1} + \)\(16\!\cdots\!80\)\( \beta_{2} - \)\(13\!\cdots\!04\)\( \beta_{3} - \)\(14\!\cdots\!36\)\( \beta_{4} - \)\(16\!\cdots\!96\)\( \beta_{5} - \)\(15\!\cdots\!64\)\( \beta_{6} + \)\(14\!\cdots\!76\)\( \beta_{7} - \)\(58\!\cdots\!60\)\( \beta_{8} - \)\(10\!\cdots\!40\)\( \beta_{9}) q^{28} +(-\)\(23\!\cdots\!70\)\( + \)\(10\!\cdots\!37\)\( \beta_{1} + \)\(55\!\cdots\!84\)\( \beta_{2} - \)\(40\!\cdots\!30\)\( \beta_{3} - \)\(68\!\cdots\!21\)\( \beta_{4} - \)\(13\!\cdots\!16\)\( \beta_{5} - \)\(64\!\cdots\!24\)\( \beta_{6} + \)\(46\!\cdots\!08\)\( \beta_{7} + \)\(19\!\cdots\!48\)\( \beta_{8} + \)\(10\!\cdots\!80\)\( \beta_{9}) q^{29} +(\)\(21\!\cdots\!08\)\( - \)\(15\!\cdots\!80\)\( \beta_{1} + \)\(29\!\cdots\!14\)\( \beta_{2} + \)\(98\!\cdots\!28\)\( \beta_{3} + \)\(64\!\cdots\!98\)\( \beta_{4} + \)\(17\!\cdots\!50\)\( \beta_{5} + \)\(14\!\cdots\!80\)\( \beta_{6} - \)\(63\!\cdots\!60\)\( \beta_{7} - \)\(24\!\cdots\!60\)\( \beta_{8} - \)\(85\!\cdots\!60\)\( \beta_{9}) q^{30} +(\)\(16\!\cdots\!92\)\( + \)\(16\!\cdots\!76\)\( \beta_{1} + \)\(63\!\cdots\!32\)\( \beta_{2} + \)\(22\!\cdots\!12\)\( \beta_{3} + \)\(45\!\cdots\!28\)\( \beta_{4} - \)\(25\!\cdots\!44\)\( \beta_{5} - \)\(88\!\cdots\!28\)\( \beta_{6} + \)\(54\!\cdots\!76\)\( \beta_{7} + \)\(21\!\cdots\!56\)\( \beta_{8} + \)\(67\!\cdots\!60\)\( \beta_{9}) q^{31} +(-\)\(83\!\cdots\!60\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} + \)\(17\!\cdots\!40\)\( \beta_{2} - \)\(13\!\cdots\!72\)\( \beta_{3} - \)\(43\!\cdots\!60\)\( \beta_{4} - \)\(71\!\cdots\!72\)\( \beta_{5} + \)\(22\!\cdots\!96\)\( \beta_{6} - \)\(32\!\cdots\!80\)\( \beta_{7} - \)\(15\!\cdots\!40\)\( \beta_{8} - \)\(49\!\cdots\!60\)\( \beta_{9}) q^{32} +(-\)\(35\!\cdots\!20\)\( + \)\(39\!\cdots\!30\)\( \beta_{1} + \)\(51\!\cdots\!86\)\( \beta_{2} - \)\(96\!\cdots\!27\)\( \beta_{3} - \)\(72\!\cdots\!96\)\( \beta_{4} + \)\(49\!\cdots\!56\)\( \beta_{5} + \)\(40\!\cdots\!35\)\( \beta_{6} + \)\(12\!\cdots\!81\)\( \beta_{7} + \)\(82\!\cdots\!15\)\( \beta_{8} + \)\(33\!\cdots\!60\)\( \beta_{9}) q^{33} +(-\)\(41\!\cdots\!84\)\( - \)\(91\!\cdots\!58\)\( \beta_{1} + \)\(32\!\cdots\!46\)\( \beta_{2} - \)\(54\!\cdots\!16\)\( \beta_{3} + \)\(10\!\cdots\!76\)\( \beta_{4} - \)\(12\!\cdots\!02\)\( \beta_{5} - \)\(35\!\cdots\!26\)\( \beta_{6} - \)\(14\!\cdots\!88\)\( \beta_{7} - \)\(31\!\cdots\!08\)\( \beta_{8} - \)\(20\!\cdots\!50\)\( \beta_{9}) q^{34} +(\)\(13\!\cdots\!04\)\( - \)\(10\!\cdots\!44\)\( \beta_{1} - \)\(78\!\cdots\!52\)\( \beta_{2} + \)\(12\!\cdots\!40\)\( \beta_{3} + \)\(51\!\cdots\!00\)\( \beta_{4} - \)\(15\!\cdots\!20\)\( \beta_{5} + \)\(14\!\cdots\!68\)\( \beta_{6} - \)\(23\!\cdots\!56\)\( \beta_{7} + \)\(40\!\cdots\!24\)\( \beta_{8} + \)\(11\!\cdots\!04\)\( \beta_{9}) q^{35} +(\)\(19\!\cdots\!96\)\( + \)\(10\!\cdots\!11\)\( \beta_{1} - \)\(10\!\cdots\!74\)\( \beta_{2} + \)\(46\!\cdots\!61\)\( \beta_{3} - \)\(54\!\cdots\!40\)\( \beta_{4} + \)\(69\!\cdots\!36\)\( \beta_{5} - \)\(60\!\cdots\!60\)\( \beta_{6} + \)\(20\!\cdots\!40\)\( \beta_{7} + \)\(59\!\cdots\!60\)\( \beta_{8} - \)\(60\!\cdots\!20\)\( \beta_{9}) q^{36} +(-\)\(29\!\cdots\!90\)\( - \)\(75\!\cdots\!91\)\( \beta_{1} - \)\(42\!\cdots\!40\)\( \beta_{2} - \)\(60\!\cdots\!08\)\( \beta_{3} - \)\(21\!\cdots\!37\)\( \beta_{4} + \)\(10\!\cdots\!08\)\( \beta_{5} - \)\(26\!\cdots\!58\)\( \beta_{6} - \)\(81\!\cdots\!98\)\( \beta_{7} - \)\(63\!\cdots\!50\)\( \beta_{8} + \)\(27\!\cdots\!00\)\( \beta_{9}) q^{37} +(-\)\(34\!\cdots\!80\)\( - \)\(53\!\cdots\!06\)\( \beta_{1} - \)\(14\!\cdots\!17\)\( \beta_{2} - \)\(95\!\cdots\!72\)\( \beta_{3} + \)\(11\!\cdots\!71\)\( \beta_{4} - \)\(18\!\cdots\!95\)\( \beta_{5} + \)\(13\!\cdots\!32\)\( \beta_{6} + \)\(22\!\cdots\!04\)\( \beta_{7} + \)\(37\!\cdots\!60\)\( \beta_{8} - \)\(11\!\cdots\!60\)\( \beta_{9}) q^{38} +(-\)\(37\!\cdots\!96\)\( + \)\(79\!\cdots\!63\)\( \beta_{1} - \)\(52\!\cdots\!27\)\( \beta_{2} - \)\(88\!\cdots\!03\)\( \beta_{3} + \)\(10\!\cdots\!87\)\( \beta_{4} + \)\(34\!\cdots\!21\)\( \beta_{5} + \)\(12\!\cdots\!93\)\( \beta_{6} + \)\(20\!\cdots\!64\)\( \beta_{7} - \)\(13\!\cdots\!96\)\( \beta_{8} + \)\(37\!\cdots\!20\)\( \beta_{9}) q^{39} +(\)\(67\!\cdots\!40\)\( - \)\(10\!\cdots\!80\)\( \beta_{1} + \)\(65\!\cdots\!40\)\( \beta_{2} + \)\(20\!\cdots\!10\)\( \beta_{3} - \)\(35\!\cdots\!90\)\( \beta_{4} + \)\(23\!\cdots\!00\)\( \beta_{5} - \)\(32\!\cdots\!40\)\( \beta_{6} - \)\(16\!\cdots\!70\)\( \beta_{7} + \)\(67\!\cdots\!80\)\( \beta_{8} - \)\(92\!\cdots\!20\)\( \beta_{9}) q^{40} +(\)\(25\!\cdots\!82\)\( + \)\(28\!\cdots\!32\)\( \beta_{1} + \)\(57\!\cdots\!16\)\( \beta_{2} + \)\(30\!\cdots\!82\)\( \beta_{3} + \)\(58\!\cdots\!40\)\( \beta_{4} - \)\(10\!\cdots\!96\)\( \beta_{5} + \)\(15\!\cdots\!10\)\( \beta_{6} + \)\(73\!\cdots\!10\)\( \beta_{7} + \)\(31\!\cdots\!90\)\( \beta_{8} + \)\(49\!\cdots\!20\)\( \beta_{9}) q^{41} +(-\)\(15\!\cdots\!80\)\( - \)\(16\!\cdots\!00\)\( \beta_{1} + \)\(72\!\cdots\!44\)\( \beta_{2} - \)\(26\!\cdots\!48\)\( \beta_{3} + \)\(49\!\cdots\!00\)\( \beta_{4} - \)\(26\!\cdots\!08\)\( \beta_{5} - \)\(12\!\cdots\!76\)\( \beta_{6} - \)\(19\!\cdots\!40\)\( \beta_{7} - \)\(29\!\cdots\!20\)\( \beta_{8} + \)\(10\!\cdots\!20\)\( \beta_{9}) q^{42} +(\)\(30\!\cdots\!00\)\( - \)\(60\!\cdots\!50\)\( \beta_{1} - \)\(16\!\cdots\!17\)\( \beta_{2} - \)\(27\!\cdots\!20\)\( \beta_{3} - \)\(19\!\cdots\!64\)\( \beta_{4} + \)\(27\!\cdots\!92\)\( \beta_{5} - \)\(23\!\cdots\!24\)\( \beta_{6} + \)\(22\!\cdots\!04\)\( \beta_{7} + \)\(16\!\cdots\!00\)\( \beta_{8} - \)\(78\!\cdots\!00\)\( \beta_{9}) q^{43} +(-\)\(12\!\cdots\!04\)\( - \)\(23\!\cdots\!72\)\( \beta_{1} - \)\(13\!\cdots\!20\)\( \beta_{2} - \)\(63\!\cdots\!16\)\( \beta_{3} - \)\(59\!\cdots\!32\)\( \beta_{4} - \)\(25\!\cdots\!44\)\( \beta_{5} + \)\(13\!\cdots\!72\)\( \beta_{6} + \)\(29\!\cdots\!16\)\( \beta_{7} - \)\(67\!\cdots\!64\)\( \beta_{8} + \)\(35\!\cdots\!20\)\( \beta_{9}) q^{44} +(\)\(98\!\cdots\!58\)\( - \)\(17\!\cdots\!35\)\( \beta_{1} - \)\(53\!\cdots\!16\)\( \beta_{2} + \)\(11\!\cdots\!48\)\( \beta_{3} + \)\(26\!\cdots\!43\)\( \beta_{4} - \)\(36\!\cdots\!00\)\( \beta_{5} - \)\(22\!\cdots\!10\)\( \beta_{6} + \)\(31\!\cdots\!70\)\( \beta_{7} + \)\(19\!\cdots\!70\)\( \beta_{8} - \)\(11\!\cdots\!80\)\( \beta_{9}) q^{45} +(\)\(75\!\cdots\!52\)\( - \)\(87\!\cdots\!68\)\( \beta_{1} + \)\(13\!\cdots\!06\)\( \beta_{2} + \)\(33\!\cdots\!72\)\( \beta_{3} + \)\(59\!\cdots\!90\)\( \beta_{4} + \)\(10\!\cdots\!74\)\( \beta_{5} - \)\(65\!\cdots\!00\)\( \beta_{6} - \)\(54\!\cdots\!80\)\( \beta_{7} - \)\(27\!\cdots\!60\)\( \beta_{8} + \)\(19\!\cdots\!80\)\( \beta_{9}) q^{46} +(-\)\(19\!\cdots\!20\)\( - \)\(17\!\cdots\!02\)\( \beta_{1} + \)\(26\!\cdots\!06\)\( \beta_{2} - \)\(11\!\cdots\!30\)\( \beta_{3} + \)\(17\!\cdots\!82\)\( \beta_{4} + \)\(36\!\cdots\!54\)\( \beta_{5} + \)\(38\!\cdots\!62\)\( \beta_{6} + \)\(35\!\cdots\!28\)\( \beta_{7} - \)\(10\!\cdots\!40\)\( \beta_{8} + \)\(28\!\cdots\!40\)\( \beta_{9}) q^{47} +(-\)\(94\!\cdots\!80\)\( - \)\(20\!\cdots\!76\)\( \beta_{1} + \)\(33\!\cdots\!24\)\( \beta_{2} - \)\(70\!\cdots\!72\)\( \beta_{3} - \)\(16\!\cdots\!00\)\( \beta_{4} - \)\(27\!\cdots\!92\)\( \beta_{5} + \)\(19\!\cdots\!76\)\( \beta_{6} - \)\(13\!\cdots\!20\)\( \beta_{7} + \)\(92\!\cdots\!00\)\( \beta_{8} - \)\(40\!\cdots\!00\)\( \beta_{9}) q^{48} +(\)\(44\!\cdots\!13\)\( - \)\(16\!\cdots\!48\)\( \beta_{1} - \)\(28\!\cdots\!84\)\( \beta_{2} + \)\(10\!\cdots\!32\)\( \beta_{3} + \)\(45\!\cdots\!60\)\( \beta_{4} + \)\(27\!\cdots\!44\)\( \beta_{5} - \)\(79\!\cdots\!20\)\( \beta_{6} + \)\(34\!\cdots\!60\)\( \beta_{7} - \)\(37\!\cdots\!20\)\( \beta_{8} + \)\(18\!\cdots\!80\)\( \beta_{9}) q^{49} +(\)\(68\!\cdots\!60\)\( - \)\(91\!\cdots\!55\)\( \beta_{1} - \)\(19\!\cdots\!00\)\( \beta_{2} + \)\(14\!\cdots\!80\)\( \beta_{3} + \)\(51\!\cdots\!80\)\( \beta_{4} + \)\(21\!\cdots\!00\)\( \beta_{5} + \)\(33\!\cdots\!60\)\( \beta_{6} - \)\(34\!\cdots\!20\)\( \beta_{7} + \)\(94\!\cdots\!80\)\( \beta_{8} - \)\(53\!\cdots\!20\)\( \beta_{9}) q^{50} +(-\)\(32\!\cdots\!28\)\( - \)\(23\!\cdots\!34\)\( \beta_{1} - \)\(71\!\cdots\!24\)\( \beta_{2} - \)\(11\!\cdots\!26\)\( \beta_{3} - \)\(27\!\cdots\!26\)\( \beta_{4} - \)\(66\!\cdots\!98\)\( \beta_{5} - \)\(60\!\cdots\!74\)\( \beta_{6} - \)\(10\!\cdots\!12\)\( \beta_{7} - \)\(71\!\cdots\!92\)\( \beta_{8} + \)\(65\!\cdots\!00\)\( \beta_{9}) q^{51} +(-\)\(36\!\cdots\!00\)\( + \)\(88\!\cdots\!50\)\( \beta_{1} - \)\(95\!\cdots\!52\)\( \beta_{2} - \)\(98\!\cdots\!90\)\( \beta_{3} - \)\(40\!\cdots\!68\)\( \beta_{4} - \)\(76\!\cdots\!96\)\( \beta_{5} + \)\(21\!\cdots\!12\)\( \beta_{6} + \)\(47\!\cdots\!48\)\( \beta_{7} - \)\(59\!\cdots\!00\)\( \beta_{8} + \)\(25\!\cdots\!00\)\( \beta_{9}) q^{52} +(-\)\(20\!\cdots\!10\)\( + \)\(90\!\cdots\!85\)\( \beta_{1} + \)\(67\!\cdots\!72\)\( \beta_{2} + \)\(11\!\cdots\!24\)\( \beta_{3} - \)\(45\!\cdots\!69\)\( \beta_{4} + \)\(53\!\cdots\!56\)\( \beta_{5} - \)\(25\!\cdots\!26\)\( \beta_{6} - \)\(44\!\cdots\!46\)\( \beta_{7} + \)\(33\!\cdots\!10\)\( \beta_{8} - \)\(19\!\cdots\!60\)\( \beta_{9}) q^{53} +(-\)\(19\!\cdots\!20\)\( + \)\(87\!\cdots\!80\)\( \beta_{1} + \)\(25\!\cdots\!58\)\( \beta_{2} + \)\(53\!\cdots\!36\)\( \beta_{3} + \)\(16\!\cdots\!38\)\( \beta_{4} + \)\(93\!\cdots\!66\)\( \beta_{5} + \)\(32\!\cdots\!12\)\( \beta_{6} - \)\(14\!\cdots\!04\)\( \beta_{7} - \)\(93\!\cdots\!24\)\( \beta_{8} + \)\(66\!\cdots\!60\)\( \beta_{9}) q^{54} +(\)\(30\!\cdots\!28\)\( + \)\(19\!\cdots\!85\)\( \beta_{1} + \)\(12\!\cdots\!39\)\( \beta_{2} + \)\(17\!\cdots\!63\)\( \beta_{3} - \)\(38\!\cdots\!67\)\( \beta_{4} - \)\(95\!\cdots\!25\)\( \beta_{5} - \)\(14\!\cdots\!25\)\( \beta_{6} - \)\(12\!\cdots\!00\)\( \beta_{7} + \)\(11\!\cdots\!00\)\( \beta_{8} - \)\(11\!\cdots\!00\)\( \beta_{9}) q^{55} +(\)\(12\!\cdots\!80\)\( + \)\(10\!\cdots\!08\)\( \beta_{1} - \)\(18\!\cdots\!72\)\( \beta_{2} + \)\(37\!\cdots\!20\)\( \beta_{3} - \)\(55\!\cdots\!24\)\( \beta_{4} + \)\(16\!\cdots\!72\)\( \beta_{5} + \)\(30\!\cdots\!24\)\( \beta_{6} + \)\(55\!\cdots\!92\)\( \beta_{7} + \)\(28\!\cdots\!52\)\( \beta_{8} - \)\(11\!\cdots\!80\)\( \beta_{9}) q^{56} +(-\)\(14\!\cdots\!60\)\( + \)\(85\!\cdots\!34\)\( \beta_{1} - \)\(66\!\cdots\!10\)\( \beta_{2} - \)\(20\!\cdots\!65\)\( \beta_{3} + \)\(17\!\cdots\!36\)\( \beta_{4} + \)\(11\!\cdots\!92\)\( \beta_{5} - \)\(11\!\cdots\!99\)\( \beta_{6} - \)\(26\!\cdots\!81\)\( \beta_{7} - \)\(19\!\cdots\!95\)\( \beta_{8} + \)\(15\!\cdots\!20\)\( \beta_{9}) q^{57} +(-\)\(13\!\cdots\!20\)\( + \)\(30\!\cdots\!40\)\( \beta_{1} - \)\(15\!\cdots\!69\)\( \beta_{2} - \)\(15\!\cdots\!02\)\( \beta_{3} + \)\(46\!\cdots\!38\)\( \beta_{4} + \)\(77\!\cdots\!79\)\( \beta_{5} - \)\(75\!\cdots\!71\)\( \beta_{6} + \)\(55\!\cdots\!22\)\( \beta_{7} + \)\(49\!\cdots\!70\)\( \beta_{8} - \)\(58\!\cdots\!45\)\( \beta_{9}) q^{58} +(\)\(32\!\cdots\!80\)\( - \)\(85\!\cdots\!82\)\( \beta_{1} - \)\(18\!\cdots\!33\)\( \beta_{2} + \)\(14\!\cdots\!32\)\( \beta_{3} - \)\(41\!\cdots\!68\)\( \beta_{4} - \)\(55\!\cdots\!72\)\( \beta_{5} - \)\(31\!\cdots\!72\)\( \beta_{6} + \)\(39\!\cdots\!64\)\( \beta_{7} - \)\(49\!\cdots\!76\)\( \beta_{8} + \)\(10\!\cdots\!00\)\( \beta_{9}) q^{59} +(\)\(21\!\cdots\!64\)\( - \)\(60\!\cdots\!44\)\( \beta_{1} + \)\(20\!\cdots\!28\)\( \beta_{2} + \)\(70\!\cdots\!00\)\( \beta_{3} - \)\(66\!\cdots\!40\)\( \beta_{4} + \)\(16\!\cdots\!80\)\( \beta_{5} + \)\(93\!\cdots\!68\)\( \beta_{6} - \)\(61\!\cdots\!56\)\( \beta_{7} - \)\(12\!\cdots\!76\)\( \beta_{8} + \)\(70\!\cdots\!04\)\( \beta_{9}) q^{60} +(\)\(48\!\cdots\!42\)\( - \)\(67\!\cdots\!23\)\( \beta_{1} - \)\(15\!\cdots\!32\)\( \beta_{2} - \)\(53\!\cdots\!60\)\( \beta_{3} + \)\(80\!\cdots\!99\)\( \beta_{4} - \)\(92\!\cdots\!84\)\( \beta_{5} - \)\(11\!\cdots\!14\)\( \beta_{6} + \)\(18\!\cdots\!58\)\( \beta_{7} + \)\(64\!\cdots\!18\)\( \beta_{8} - \)\(11\!\cdots\!40\)\( \beta_{9}) q^{61} +(-\)\(16\!\cdots\!20\)\( - \)\(30\!\cdots\!44\)\( \beta_{1} + \)\(16\!\cdots\!20\)\( \beta_{2} - \)\(37\!\cdots\!88\)\( \beta_{3} + \)\(36\!\cdots\!00\)\( \beta_{4} - \)\(64\!\cdots\!28\)\( \beta_{5} - \)\(84\!\cdots\!16\)\( \beta_{6} - \)\(17\!\cdots\!00\)\( \beta_{7} - \)\(11\!\cdots\!40\)\( \beta_{8} + \)\(39\!\cdots\!40\)\( \beta_{9}) q^{62} +(-\)\(29\!\cdots\!40\)\( - \)\(51\!\cdots\!27\)\( \beta_{1} - \)\(17\!\cdots\!25\)\( \beta_{2} - \)\(35\!\cdots\!57\)\( \beta_{3} - \)\(11\!\cdots\!23\)\( \beta_{4} - \)\(25\!\cdots\!33\)\( \beta_{5} + \)\(37\!\cdots\!63\)\( \beta_{6} - \)\(43\!\cdots\!72\)\( \beta_{7} + \)\(34\!\cdots\!40\)\( \beta_{8} - \)\(58\!\cdots\!40\)\( \beta_{9}) q^{63} +(\)\(12\!\cdots\!48\)\( + \)\(16\!\cdots\!96\)\( \beta_{1} - \)\(53\!\cdots\!60\)\( \beta_{2} + \)\(34\!\cdots\!76\)\( \beta_{3} - \)\(33\!\cdots\!20\)\( \beta_{4} + \)\(11\!\cdots\!68\)\( \beta_{5} - \)\(58\!\cdots\!00\)\( \beta_{6} + \)\(17\!\cdots\!40\)\( \beta_{7} + \)\(25\!\cdots\!80\)\( \beta_{8} - \)\(79\!\cdots\!40\)\( \beta_{9}) q^{64} +(\)\(27\!\cdots\!08\)\( + \)\(27\!\cdots\!72\)\( \beta_{1} - \)\(25\!\cdots\!44\)\( \beta_{2} - \)\(35\!\cdots\!10\)\( \beta_{3} + \)\(17\!\cdots\!60\)\( \beta_{4} - \)\(16\!\cdots\!40\)\( \beta_{5} - \)\(31\!\cdots\!34\)\( \beta_{6} - \)\(12\!\cdots\!22\)\( \beta_{7} - \)\(13\!\cdots\!62\)\( \beta_{8} + \)\(72\!\cdots\!48\)\( \beta_{9}) q^{65} +(-\)\(45\!\cdots\!16\)\( + \)\(10\!\cdots\!68\)\( \beta_{1} + \)\(42\!\cdots\!58\)\( \beta_{2} - \)\(31\!\cdots\!48\)\( \beta_{3} + \)\(18\!\cdots\!52\)\( \beta_{4} + \)\(36\!\cdots\!26\)\( \beta_{5} + \)\(49\!\cdots\!98\)\( \beta_{6} - \)\(38\!\cdots\!76\)\( \beta_{7} - \)\(14\!\cdots\!16\)\( \beta_{8} - \)\(19\!\cdots\!50\)\( \beta_{9}) q^{66} +(-\)\(14\!\cdots\!80\)\( + \)\(20\!\cdots\!64\)\( \beta_{1} + \)\(69\!\cdots\!59\)\( \beta_{2} - \)\(33\!\cdots\!62\)\( \beta_{3} - \)\(37\!\cdots\!94\)\( \beta_{4} - \)\(45\!\cdots\!70\)\( \beta_{5} + \)\(35\!\cdots\!02\)\( \beta_{6} - \)\(87\!\cdots\!16\)\( \beta_{7} + \)\(63\!\cdots\!40\)\( \beta_{8} + \)\(18\!\cdots\!60\)\( \beta_{9}) q^{67} +(\)\(11\!\cdots\!40\)\( + \)\(62\!\cdots\!58\)\( \beta_{1} - \)\(53\!\cdots\!44\)\( \beta_{2} + \)\(33\!\cdots\!46\)\( \beta_{3} - \)\(67\!\cdots\!56\)\( \beta_{4} + \)\(28\!\cdots\!24\)\( \beta_{5} - \)\(38\!\cdots\!64\)\( \beta_{6} + \)\(65\!\cdots\!16\)\( \beta_{7} + \)\(23\!\cdots\!80\)\( \beta_{8} + \)\(72\!\cdots\!20\)\( \beta_{9}) q^{68} +(\)\(26\!\cdots\!44\)\( - \)\(12\!\cdots\!72\)\( \beta_{1} - \)\(78\!\cdots\!36\)\( \beta_{2} + \)\(81\!\cdots\!22\)\( \beta_{3} - \)\(24\!\cdots\!28\)\( \beta_{4} - \)\(35\!\cdots\!76\)\( \beta_{5} - \)\(53\!\cdots\!42\)\( \beta_{6} - \)\(18\!\cdots\!06\)\( \beta_{7} - \)\(94\!\cdots\!06\)\( \beta_{8} - \)\(35\!\cdots\!40\)\( \beta_{9}) q^{69} +(\)\(12\!\cdots\!24\)\( - \)\(23\!\cdots\!00\)\( \beta_{1} - \)\(14\!\cdots\!68\)\( \beta_{2} + \)\(91\!\cdots\!24\)\( \beta_{3} + \)\(11\!\cdots\!84\)\( \beta_{4} - \)\(91\!\cdots\!00\)\( \beta_{5} + \)\(21\!\cdots\!60\)\( \beta_{6} - \)\(39\!\cdots\!20\)\( \beta_{7} + \)\(11\!\cdots\!80\)\( \beta_{8} + \)\(69\!\cdots\!80\)\( \beta_{9}) q^{70} +(\)\(16\!\cdots\!32\)\( - \)\(28\!\cdots\!99\)\( \beta_{1} - \)\(20\!\cdots\!81\)\( \beta_{2} + \)\(72\!\cdots\!15\)\( \beta_{3} - \)\(50\!\cdots\!63\)\( \beta_{4} + \)\(26\!\cdots\!63\)\( \beta_{5} - \)\(25\!\cdots\!37\)\( \beta_{6} + \)\(13\!\cdots\!84\)\( \beta_{7} + \)\(34\!\cdots\!84\)\( \beta_{8} + \)\(32\!\cdots\!60\)\( \beta_{9}) q^{71} +(-\)\(12\!\cdots\!40\)\( - \)\(34\!\cdots\!28\)\( \beta_{1} + \)\(11\!\cdots\!46\)\( \beta_{2} - \)\(18\!\cdots\!91\)\( \beta_{3} - \)\(10\!\cdots\!55\)\( \beta_{4} - \)\(13\!\cdots\!76\)\( \beta_{5} - \)\(37\!\cdots\!82\)\( \beta_{6} - \)\(38\!\cdots\!85\)\( \beta_{7} - \)\(18\!\cdots\!60\)\( \beta_{8} - \)\(42\!\cdots\!40\)\( \beta_{9}) q^{72} +(\)\(12\!\cdots\!70\)\( - \)\(12\!\cdots\!06\)\( \beta_{1} + \)\(47\!\cdots\!70\)\( \beta_{2} + \)\(25\!\cdots\!63\)\( \beta_{3} - \)\(91\!\cdots\!64\)\( \beta_{4} + \)\(63\!\cdots\!20\)\( \beta_{5} + \)\(13\!\cdots\!17\)\( \beta_{6} + \)\(28\!\cdots\!39\)\( \beta_{7} + \)\(31\!\cdots\!85\)\( \beta_{8} + \)\(12\!\cdots\!40\)\( \beta_{9}) q^{73} +(\)\(54\!\cdots\!16\)\( + \)\(56\!\cdots\!76\)\( \beta_{1} - \)\(54\!\cdots\!89\)\( \beta_{2} - \)\(18\!\cdots\!02\)\( \beta_{3} + \)\(29\!\cdots\!46\)\( \beta_{4} - \)\(44\!\cdots\!65\)\( \beta_{5} - \)\(57\!\cdots\!91\)\( \beta_{6} + \)\(97\!\cdots\!62\)\( \beta_{7} + \)\(16\!\cdots\!62\)\( \beta_{8} - \)\(13\!\cdots\!45\)\( \beta_{9}) q^{74} +(\)\(54\!\cdots\!20\)\( + \)\(11\!\cdots\!90\)\( \beta_{1} + \)\(29\!\cdots\!75\)\( \beta_{2} + \)\(33\!\cdots\!60\)\( \beta_{3} - \)\(21\!\cdots\!40\)\( \beta_{4} + \)\(78\!\cdots\!00\)\( \beta_{5} - \)\(79\!\cdots\!80\)\( \beta_{6} - \)\(31\!\cdots\!40\)\( \beta_{7} - \)\(19\!\cdots\!40\)\( \beta_{8} - \)\(31\!\cdots\!40\)\( \beta_{9}) q^{75} +(-\)\(19\!\cdots\!40\)\( + \)\(70\!\cdots\!96\)\( \beta_{1} - \)\(22\!\cdots\!44\)\( \beta_{2} - \)\(23\!\cdots\!60\)\( \beta_{3} + \)\(61\!\cdots\!92\)\( \beta_{4} + \)\(74\!\cdots\!44\)\( \beta_{5} - \)\(10\!\cdots\!52\)\( \beta_{6} + \)\(36\!\cdots\!24\)\( \beta_{7} + \)\(37\!\cdots\!84\)\( \beta_{8} + \)\(15\!\cdots\!00\)\( \beta_{9}) q^{76} +(-\)\(16\!\cdots\!00\)\( + \)\(60\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2} - \)\(47\!\cdots\!42\)\( \beta_{3} - \)\(33\!\cdots\!60\)\( \beta_{4} - \)\(44\!\cdots\!32\)\( \beta_{5} + \)\(36\!\cdots\!26\)\( \beta_{6} - \)\(17\!\cdots\!10\)\( \beta_{7} - \)\(42\!\cdots\!50\)\( \beta_{8} - \)\(27\!\cdots\!00\)\( \beta_{9}) q^{77} +(-\)\(89\!\cdots\!00\)\( + \)\(82\!\cdots\!44\)\( \beta_{1} + \)\(19\!\cdots\!46\)\( \beta_{2} - \)\(21\!\cdots\!64\)\( \beta_{3} + \)\(49\!\cdots\!62\)\( \beta_{4} - \)\(36\!\cdots\!50\)\( \beta_{5} - \)\(17\!\cdots\!16\)\( \beta_{6} + \)\(47\!\cdots\!28\)\( \beta_{7} - \)\(11\!\cdots\!00\)\( \beta_{8} - \)\(63\!\cdots\!00\)\( \beta_{9}) q^{78} +(\)\(15\!\cdots\!60\)\( - \)\(26\!\cdots\!38\)\( \beta_{1} + \)\(85\!\cdots\!26\)\( \beta_{2} + \)\(71\!\cdots\!18\)\( \beta_{3} - \)\(13\!\cdots\!82\)\( \beta_{4} - \)\(19\!\cdots\!74\)\( \beta_{5} - \)\(10\!\cdots\!78\)\( \beta_{6} - \)\(49\!\cdots\!84\)\( \beta_{7} + \)\(18\!\cdots\!36\)\( \beta_{8} + \)\(14\!\cdots\!20\)\( \beta_{9}) q^{79} +(\)\(71\!\cdots\!24\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(43\!\cdots\!32\)\( \beta_{2} + \)\(21\!\cdots\!24\)\( \beta_{3} + \)\(10\!\cdots\!84\)\( \beta_{4} + \)\(44\!\cdots\!00\)\( \beta_{5} + \)\(14\!\cdots\!60\)\( \beta_{6} - \)\(79\!\cdots\!20\)\( \beta_{7} + \)\(82\!\cdots\!80\)\( \beta_{8} - \)\(34\!\cdots\!20\)\( \beta_{9}) q^{80} +(\)\(68\!\cdots\!21\)\( - \)\(12\!\cdots\!30\)\( \beta_{1} - \)\(63\!\cdots\!90\)\( \beta_{2} - \)\(14\!\cdots\!71\)\( \beta_{3} - \)\(28\!\cdots\!88\)\( \beta_{4} - \)\(72\!\cdots\!52\)\( \beta_{5} + \)\(37\!\cdots\!83\)\( \beta_{6} + \)\(33\!\cdots\!89\)\( \beta_{7} - \)\(28\!\cdots\!41\)\( \beta_{8} + \)\(19\!\cdots\!40\)\( \beta_{9}) q^{81} +(-\)\(81\!\cdots\!20\)\( - \)\(43\!\cdots\!94\)\( \beta_{1} - \)\(54\!\cdots\!32\)\( \beta_{2} - \)\(33\!\cdots\!68\)\( \beta_{3} - \)\(19\!\cdots\!92\)\( \beta_{4} - \)\(12\!\cdots\!92\)\( \beta_{5} + \)\(77\!\cdots\!32\)\( \beta_{6} - \)\(44\!\cdots\!28\)\( \beta_{7} - \)\(14\!\cdots\!20\)\( \beta_{8} + \)\(98\!\cdots\!20\)\( \beta_{9}) q^{82} +(-\)\(12\!\cdots\!40\)\( - \)\(40\!\cdots\!94\)\( \beta_{1} - \)\(50\!\cdots\!27\)\( \beta_{2} - \)\(81\!\cdots\!60\)\( \beta_{3} - \)\(50\!\cdots\!20\)\( \beta_{4} - \)\(83\!\cdots\!40\)\( \beta_{5} - \)\(41\!\cdots\!20\)\( \beta_{6} - \)\(59\!\cdots\!60\)\( \beta_{7} + \)\(47\!\cdots\!40\)\( \beta_{8} - \)\(31\!\cdots\!40\)\( \beta_{9}) q^{83} +(\)\(14\!\cdots\!96\)\( + \)\(27\!\cdots\!00\)\( \beta_{1} - \)\(54\!\cdots\!84\)\( \beta_{2} + \)\(19\!\cdots\!00\)\( \beta_{3} + \)\(30\!\cdots\!00\)\( \beta_{4} + \)\(73\!\cdots\!68\)\( \beta_{5} + \)\(41\!\cdots\!60\)\( \beta_{6} + \)\(29\!\cdots\!00\)\( \beta_{7} + \)\(22\!\cdots\!20\)\( \beta_{8} + \)\(33\!\cdots\!80\)\( \beta_{9}) q^{84} +(\)\(14\!\cdots\!04\)\( + \)\(24\!\cdots\!46\)\( \beta_{1} + \)\(93\!\cdots\!88\)\( \beta_{2} + \)\(11\!\cdots\!30\)\( \beta_{3} + \)\(41\!\cdots\!90\)\( \beta_{4} - \)\(34\!\cdots\!20\)\( \beta_{5} + \)\(12\!\cdots\!38\)\( \beta_{6} - \)\(31\!\cdots\!46\)\( \beta_{7} - \)\(38\!\cdots\!66\)\( \beta_{8} + \)\(43\!\cdots\!64\)\( \beta_{9}) q^{85} +(\)\(66\!\cdots\!92\)\( + \)\(17\!\cdots\!74\)\( \beta_{1} + \)\(18\!\cdots\!25\)\( \beta_{2} + \)\(35\!\cdots\!12\)\( \beta_{3} - \)\(26\!\cdots\!31\)\( \beta_{4} - \)\(31\!\cdots\!97\)\( \beta_{5} - \)\(37\!\cdots\!84\)\( \beta_{6} - \)\(23\!\cdots\!12\)\( \beta_{7} + \)\(71\!\cdots\!88\)\( \beta_{8} - \)\(20\!\cdots\!80\)\( \beta_{9}) q^{86} +(\)\(42\!\cdots\!60\)\( + \)\(12\!\cdots\!99\)\( \beta_{1} + \)\(17\!\cdots\!37\)\( \beta_{2} - \)\(96\!\cdots\!43\)\( \beta_{3} + \)\(16\!\cdots\!31\)\( \beta_{4} + \)\(44\!\cdots\!09\)\( \beta_{5} + \)\(21\!\cdots\!65\)\( \beta_{6} + \)\(57\!\cdots\!64\)\( \beta_{7} + \)\(45\!\cdots\!20\)\( \beta_{8} + \)\(29\!\cdots\!80\)\( \beta_{9}) q^{87} +(\)\(13\!\cdots\!60\)\( + \)\(63\!\cdots\!48\)\( \beta_{1} - \)\(23\!\cdots\!04\)\( \beta_{2} - \)\(12\!\cdots\!12\)\( \beta_{3} + \)\(19\!\cdots\!80\)\( \beta_{4} + \)\(15\!\cdots\!68\)\( \beta_{5} + \)\(16\!\cdots\!56\)\( \beta_{6} + \)\(74\!\cdots\!80\)\( \beta_{7} - \)\(40\!\cdots\!40\)\( \beta_{8} + \)\(56\!\cdots\!40\)\( \beta_{9}) q^{88} +(\)\(41\!\cdots\!90\)\( - \)\(31\!\cdots\!02\)\( \beta_{1} - \)\(95\!\cdots\!30\)\( \beta_{2} - \)\(18\!\cdots\!05\)\( \beta_{3} + \)\(94\!\cdots\!16\)\( \beta_{4} + \)\(10\!\cdots\!68\)\( \beta_{5} + \)\(71\!\cdots\!09\)\( \beta_{6} - \)\(85\!\cdots\!33\)\( \beta_{7} + \)\(51\!\cdots\!97\)\( \beta_{8} - \)\(92\!\cdots\!00\)\( \beta_{9}) q^{89} +(\)\(19\!\cdots\!48\)\( - \)\(16\!\cdots\!48\)\( \beta_{1} - \)\(33\!\cdots\!69\)\( \beta_{2} + \)\(84\!\cdots\!10\)\( \beta_{3} - \)\(25\!\cdots\!70\)\( \beta_{4} - \)\(54\!\cdots\!65\)\( \beta_{5} - \)\(13\!\cdots\!19\)\( \beta_{6} - \)\(79\!\cdots\!02\)\( \beta_{7} + \)\(72\!\cdots\!58\)\( \beta_{8} + \)\(15\!\cdots\!43\)\( \beta_{9}) q^{90} +(\)\(21\!\cdots\!92\)\( - \)\(77\!\cdots\!52\)\( \beta_{1} - \)\(33\!\cdots\!72\)\( \beta_{2} - \)\(53\!\cdots\!80\)\( \beta_{3} + \)\(40\!\cdots\!16\)\( \beta_{4} + \)\(19\!\cdots\!52\)\( \beta_{5} - \)\(58\!\cdots\!36\)\( \beta_{6} + \)\(19\!\cdots\!92\)\( \beta_{7} - \)\(27\!\cdots\!68\)\( \beta_{8} - \)\(81\!\cdots\!60\)\( \beta_{9}) q^{91} +(\)\(76\!\cdots\!20\)\( - \)\(11\!\cdots\!40\)\( \beta_{1} + \)\(90\!\cdots\!16\)\( \beta_{2} + \)\(70\!\cdots\!52\)\( \beta_{3} - \)\(93\!\cdots\!12\)\( \beta_{4} + \)\(14\!\cdots\!68\)\( \beta_{5} + \)\(18\!\cdots\!12\)\( \beta_{6} - \)\(22\!\cdots\!48\)\( \beta_{7} + \)\(57\!\cdots\!00\)\( \beta_{8} - \)\(10\!\cdots\!00\)\( \beta_{9}) q^{92} +(\)\(41\!\cdots\!80\)\( - \)\(19\!\cdots\!76\)\( \beta_{1} + \)\(26\!\cdots\!00\)\( \beta_{2} - \)\(15\!\cdots\!76\)\( \beta_{3} + \)\(54\!\cdots\!28\)\( \beta_{4} - \)\(11\!\cdots\!20\)\( \beta_{5} - \)\(83\!\cdots\!44\)\( \beta_{6} - \)\(43\!\cdots\!88\)\( \beta_{7} + \)\(98\!\cdots\!60\)\( \beta_{8} + \)\(25\!\cdots\!40\)\( \beta_{9}) q^{93} +(\)\(18\!\cdots\!76\)\( + \)\(12\!\cdots\!80\)\( \beta_{1} + \)\(14\!\cdots\!48\)\( \beta_{2} + \)\(23\!\cdots\!76\)\( \beta_{3} + \)\(17\!\cdots\!28\)\( \beta_{4} - \)\(58\!\cdots\!04\)\( \beta_{5} - \)\(21\!\cdots\!68\)\( \beta_{6} + \)\(24\!\cdots\!16\)\( \beta_{7} - \)\(12\!\cdots\!44\)\( \beta_{8} - \)\(69\!\cdots\!00\)\( \beta_{9}) q^{94} +(\)\(24\!\cdots\!60\)\( + \)\(26\!\cdots\!35\)\( \beta_{1} - \)\(55\!\cdots\!35\)\( \beta_{2} - \)\(41\!\cdots\!55\)\( \beta_{3} - \)\(29\!\cdots\!05\)\( \beta_{4} - \)\(87\!\cdots\!75\)\( \beta_{5} - \)\(10\!\cdots\!95\)\( \beta_{6} + \)\(77\!\cdots\!40\)\( \beta_{7} - \)\(28\!\cdots\!60\)\( \beta_{8} + \)\(23\!\cdots\!40\)\( \beta_{9}) q^{95} +(\)\(14\!\cdots\!32\)\( + \)\(62\!\cdots\!08\)\( \beta_{1} - \)\(23\!\cdots\!72\)\( \beta_{2} + \)\(16\!\cdots\!08\)\( \beta_{3} - \)\(48\!\cdots\!00\)\( \beta_{4} + \)\(22\!\cdots\!88\)\( \beta_{5} + \)\(54\!\cdots\!60\)\( \beta_{6} + \)\(40\!\cdots\!00\)\( \beta_{7} + \)\(97\!\cdots\!20\)\( \beta_{8} - \)\(39\!\cdots\!20\)\( \beta_{9}) q^{96} +(\)\(10\!\cdots\!30\)\( - \)\(41\!\cdots\!78\)\( \beta_{1} + \)\(10\!\cdots\!18\)\( \beta_{2} - \)\(35\!\cdots\!09\)\( \beta_{3} - \)\(93\!\cdots\!96\)\( \beta_{4} + \)\(72\!\cdots\!64\)\( \beta_{5} + \)\(25\!\cdots\!61\)\( \beta_{6} - \)\(16\!\cdots\!69\)\( \beta_{7} - \)\(27\!\cdots\!95\)\( \beta_{8} - \)\(24\!\cdots\!80\)\( \beta_{9}) q^{97} +(\)\(17\!\cdots\!20\)\( + \)\(60\!\cdots\!55\)\( \beta_{1} + \)\(57\!\cdots\!48\)\( \beta_{2} - \)\(10\!\cdots\!68\)\( \beta_{3} + \)\(63\!\cdots\!48\)\( \beta_{4} - \)\(32\!\cdots\!72\)\( \beta_{5} - \)\(76\!\cdots\!48\)\( \beta_{6} + \)\(81\!\cdots\!12\)\( \beta_{7} - \)\(33\!\cdots\!60\)\( \beta_{8} + \)\(24\!\cdots\!60\)\( \beta_{9}) q^{98} +(\)\(32\!\cdots\!84\)\( - \)\(14\!\cdots\!78\)\( \beta_{1} + \)\(43\!\cdots\!67\)\( \beta_{2} - \)\(48\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!76\)\( \beta_{4} - \)\(10\!\cdots\!72\)\( \beta_{5} + \)\(11\!\cdots\!16\)\( \beta_{6} + \)\(62\!\cdots\!68\)\( \beta_{7} + \)\(62\!\cdots\!48\)\( \beta_{8} - \)\(42\!\cdots\!60\)\( \beta_{9}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 919932722605769400q^{2} - \)\(95\!\cdots\!00\)\(q^{3} + \)\(42\!\cdots\!80\)\(q^{4} + \)\(60\!\cdots\!40\)\(q^{5} + \)\(36\!\cdots\!20\)\(q^{6} - \)\(21\!\cdots\!00\)\(q^{7} - \)\(56\!\cdots\!00\)\(q^{8} + \)\(18\!\cdots\!70\)\(q^{9} + O(q^{10}) \) \( 10q + 919932722605769400q^{2} - \)\(95\!\cdots\!00\)\(q^{3} + \)\(42\!\cdots\!80\)\(q^{4} + \)\(60\!\cdots\!40\)\(q^{5} + \)\(36\!\cdots\!20\)\(q^{6} - \)\(21\!\cdots\!00\)\(q^{7} - \)\(56\!\cdots\!00\)\(q^{8} + \)\(18\!\cdots\!70\)\(q^{9} + \)\(63\!\cdots\!40\)\(q^{10} + \)\(10\!\cdots\!20\)\(q^{11} - \)\(38\!\cdots\!00\)\(q^{12} - \)\(11\!\cdots\!00\)\(q^{13} + \)\(12\!\cdots\!60\)\(q^{14} - \)\(37\!\cdots\!20\)\(q^{15} + \)\(16\!\cdots\!60\)\(q^{16} - \)\(78\!\cdots\!00\)\(q^{17} + \)\(33\!\cdots\!00\)\(q^{18} + \)\(69\!\cdots\!00\)\(q^{19} + \)\(73\!\cdots\!20\)\(q^{20} + \)\(29\!\cdots\!20\)\(q^{21} + \)\(18\!\cdots\!00\)\(q^{22} + \)\(28\!\cdots\!00\)\(q^{23} + \)\(11\!\cdots\!00\)\(q^{24} + \)\(43\!\cdots\!50\)\(q^{25} - \)\(31\!\cdots\!80\)\(q^{26} - \)\(45\!\cdots\!00\)\(q^{27} - \)\(52\!\cdots\!00\)\(q^{28} - \)\(23\!\cdots\!00\)\(q^{29} + \)\(21\!\cdots\!80\)\(q^{30} + \)\(16\!\cdots\!20\)\(q^{31} - \)\(83\!\cdots\!00\)\(q^{32} - \)\(35\!\cdots\!00\)\(q^{33} - \)\(41\!\cdots\!40\)\(q^{34} + \)\(13\!\cdots\!40\)\(q^{35} + \)\(19\!\cdots\!60\)\(q^{36} - \)\(29\!\cdots\!00\)\(q^{37} - \)\(34\!\cdots\!00\)\(q^{38} - \)\(37\!\cdots\!60\)\(q^{39} + \)\(67\!\cdots\!00\)\(q^{40} + \)\(25\!\cdots\!20\)\(q^{41} - \)\(15\!\cdots\!00\)\(q^{42} + \)\(30\!\cdots\!00\)\(q^{43} - \)\(12\!\cdots\!40\)\(q^{44} + \)\(98\!\cdots\!80\)\(q^{45} + \)\(75\!\cdots\!20\)\(q^{46} - \)\(19\!\cdots\!00\)\(q^{47} - \)\(94\!\cdots\!00\)\(q^{48} + \)\(44\!\cdots\!30\)\(q^{49} + \)\(68\!\cdots\!00\)\(q^{50} - \)\(32\!\cdots\!80\)\(q^{51} - \)\(36\!\cdots\!00\)\(q^{52} - \)\(20\!\cdots\!00\)\(q^{53} - \)\(19\!\cdots\!00\)\(q^{54} + \)\(30\!\cdots\!80\)\(q^{55} + \)\(12\!\cdots\!00\)\(q^{56} - \)\(14\!\cdots\!00\)\(q^{57} - \)\(13\!\cdots\!00\)\(q^{58} + \)\(32\!\cdots\!00\)\(q^{59} + \)\(21\!\cdots\!40\)\(q^{60} + \)\(48\!\cdots\!20\)\(q^{61} - \)\(16\!\cdots\!00\)\(q^{62} - \)\(29\!\cdots\!00\)\(q^{63} + \)\(12\!\cdots\!80\)\(q^{64} + \)\(27\!\cdots\!80\)\(q^{65} - \)\(45\!\cdots\!60\)\(q^{66} - \)\(14\!\cdots\!00\)\(q^{67} + \)\(11\!\cdots\!00\)\(q^{68} + \)\(26\!\cdots\!40\)\(q^{69} + \)\(12\!\cdots\!40\)\(q^{70} + \)\(16\!\cdots\!20\)\(q^{71} - \)\(12\!\cdots\!00\)\(q^{72} + \)\(12\!\cdots\!00\)\(q^{73} + \)\(54\!\cdots\!60\)\(q^{74} + \)\(54\!\cdots\!00\)\(q^{75} - \)\(19\!\cdots\!00\)\(q^{76} - \)\(16\!\cdots\!00\)\(q^{77} - \)\(89\!\cdots\!00\)\(q^{78} + \)\(15\!\cdots\!00\)\(q^{79} + \)\(71\!\cdots\!40\)\(q^{80} + \)\(68\!\cdots\!10\)\(q^{81} - \)\(81\!\cdots\!00\)\(q^{82} - \)\(12\!\cdots\!00\)\(q^{83} + \)\(14\!\cdots\!60\)\(q^{84} + \)\(14\!\cdots\!40\)\(q^{85} + \)\(66\!\cdots\!20\)\(q^{86} + \)\(42\!\cdots\!00\)\(q^{87} + \)\(13\!\cdots\!00\)\(q^{88} + \)\(41\!\cdots\!00\)\(q^{89} + \)\(19\!\cdots\!80\)\(q^{90} + \)\(21\!\cdots\!20\)\(q^{91} + \)\(76\!\cdots\!00\)\(q^{92} + \)\(41\!\cdots\!00\)\(q^{93} + \)\(18\!\cdots\!60\)\(q^{94} + \)\(24\!\cdots\!00\)\(q^{95} + \)\(14\!\cdots\!20\)\(q^{96} + \)\(10\!\cdots\!00\)\(q^{97} + \)\(17\!\cdots\!00\)\(q^{98} + \)\(32\!\cdots\!40\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 5 x^{9} - \)\(94\!\cdots\!00\)\( x^{8} - \)\(56\!\cdots\!20\)\( x^{7} + \)\(29\!\cdots\!10\)\( x^{6} + \)\(30\!\cdots\!94\)\( x^{5} - \)\(35\!\cdots\!20\)\( x^{4} - \)\(44\!\cdots\!80\)\( x^{3} + \)\(12\!\cdots\!25\)\( x^{2} + \)\(12\!\cdots\!95\)\( x + \)\(23\!\cdots\!68\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 12 \)
\(\beta_{2}\)\(=\)\((\)\(\)\(48\!\cdots\!81\)\( \nu^{9} - \)\(10\!\cdots\!72\)\( \nu^{8} - \)\(41\!\cdots\!08\)\( \nu^{7} + \)\(53\!\cdots\!04\)\( \nu^{6} + \)\(11\!\cdots\!94\)\( \nu^{5} - \)\(58\!\cdots\!88\)\( \nu^{4} - \)\(11\!\cdots\!96\)\( \nu^{3} - \)\(31\!\cdots\!88\)\( \nu^{2} + \)\(29\!\cdots\!05\)\( \nu + \)\(20\!\cdots\!04\)\(\)\()/ \)\(48\!\cdots\!44\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(34\!\cdots\!41\)\( \nu^{9} - \)\(72\!\cdots\!92\)\( \nu^{8} - \)\(29\!\cdots\!88\)\( \nu^{7} + \)\(38\!\cdots\!44\)\( \nu^{6} + \)\(84\!\cdots\!34\)\( \nu^{5} - \)\(42\!\cdots\!68\)\( \nu^{4} - \)\(86\!\cdots\!56\)\( \nu^{3} + \)\(24\!\cdots\!56\)\( \nu^{2} + \)\(17\!\cdots\!25\)\( \nu - \)\(72\!\cdots\!56\)\(\)\()/ \)\(80\!\cdots\!24\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(47\!\cdots\!59\)\( \nu^{9} - \)\(20\!\cdots\!84\)\( \nu^{8} - \)\(38\!\cdots\!36\)\( \nu^{7} + \)\(13\!\cdots\!76\)\( \nu^{6} + \)\(10\!\cdots\!94\)\( \nu^{5} - \)\(24\!\cdots\!28\)\( \nu^{4} - \)\(11\!\cdots\!92\)\( \nu^{3} + \)\(10\!\cdots\!12\)\( \nu^{2} + \)\(32\!\cdots\!23\)\( \nu + \)\(26\!\cdots\!72\)\(\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(20\!\cdots\!87\)\( \nu^{9} - \)\(86\!\cdots\!12\)\( \nu^{8} - \)\(15\!\cdots\!48\)\( \nu^{7} + \)\(59\!\cdots\!68\)\( \nu^{6} + \)\(39\!\cdots\!42\)\( \nu^{5} - \)\(12\!\cdots\!04\)\( \nu^{4} - \)\(35\!\cdots\!56\)\( \nu^{3} + \)\(69\!\cdots\!16\)\( \nu^{2} + \)\(77\!\cdots\!39\)\( \nu + \)\(49\!\cdots\!96\)\(\)\()/ \)\(43\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(70\!\cdots\!99\)\( \nu^{9} + \)\(15\!\cdots\!24\)\( \nu^{8} + \)\(63\!\cdots\!96\)\( \nu^{7} - \)\(10\!\cdots\!36\)\( \nu^{6} - \)\(19\!\cdots\!34\)\( \nu^{5} + \)\(22\!\cdots\!08\)\( \nu^{4} + \)\(21\!\cdots\!12\)\( \nu^{3} - \)\(15\!\cdots\!32\)\( \nu^{2} - \)\(59\!\cdots\!03\)\( \nu + \)\(14\!\cdots\!08\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(76\!\cdots\!23\)\( \nu^{9} - \)\(90\!\cdots\!48\)\( \nu^{8} - \)\(69\!\cdots\!92\)\( \nu^{7} - \)\(43\!\cdots\!28\)\( \nu^{6} + \)\(20\!\cdots\!18\)\( \nu^{5} + \)\(23\!\cdots\!84\)\( \nu^{4} - \)\(19\!\cdots\!24\)\( \nu^{3} - \)\(34\!\cdots\!36\)\( \nu^{2} - \)\(39\!\cdots\!69\)\( \nu + \)\(69\!\cdots\!84\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(16\!\cdots\!21\)\( \nu^{9} + \)\(55\!\cdots\!96\)\( \nu^{8} - \)\(14\!\cdots\!16\)\( \nu^{7} - \)\(41\!\cdots\!44\)\( \nu^{6} + \)\(15\!\cdots\!14\)\( \nu^{5} + \)\(10\!\cdots\!32\)\( \nu^{4} - \)\(33\!\cdots\!52\)\( \nu^{3} - \)\(90\!\cdots\!28\)\( \nu^{2} + \)\(19\!\cdots\!63\)\( \nu + \)\(12\!\cdots\!32\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(49\!\cdots\!01\)\( \nu^{9} - \)\(41\!\cdots\!76\)\( \nu^{8} - \)\(42\!\cdots\!04\)\( \nu^{7} + \)\(61\!\cdots\!64\)\( \nu^{6} + \)\(12\!\cdots\!66\)\( \nu^{5} + \)\(41\!\cdots\!08\)\( \nu^{4} - \)\(13\!\cdots\!88\)\( \nu^{3} - \)\(76\!\cdots\!32\)\( \nu^{2} + \)\(41\!\cdots\!97\)\( \nu + \)\(15\!\cdots\!08\)\(\)\()/ \)\(37\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 12\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 4314966 \beta_{2} + 216035480204442455 \beta_{1} + 1083421825434706698581133759471027360\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - 194 \beta_{6} + 4418884 \beta_{5} + 285635101119 \beta_{4} + 275048905921865907 \beta_{3} - 13907955504737675909846010 \beta_{2} + 1786554483299548909187068549960989240 \beta_{1} + 234057554324064619942918589963663737469490159239234304\)\()/13824\)
\(\nu^{4}\)\(=\)\((\)\(-27276333880 \beta_{9} + 39762557716528 \beta_{8} + 65021127011131829 \beta_{7} + 153363500265211376302 \beta_{6} + 2822569104990237132874492 \beta_{5} - 17482108978014033342463972917 \beta_{4} + 317285727842857180596611876327539959 \beta_{3} - 1738064963174056947421811155714695965392010 \beta_{2} + 104940489183712277132884549742697422332955248976732256 \beta_{1} + 241949014942797889553414045068447411694678682896742843840988466575840512\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(\)\(30\!\cdots\!20\)\( \beta_{9} + \)\(23\!\cdots\!60\)\( \beta_{8} + \)\(24\!\cdots\!79\)\( \beta_{7} + \)\(33\!\cdots\!62\)\( \beta_{6} + \)\(22\!\cdots\!80\)\( \beta_{5} + \)\(86\!\cdots\!01\)\( \beta_{4} + \)\(10\!\cdots\!73\)\( \beta_{3} - \)\(44\!\cdots\!66\)\( \beta_{2} + \)\(29\!\cdots\!12\)\( \beta_{1} + \)\(71\!\cdots\!24\)\(\)\()/62208\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(18\!\cdots\!60\)\( \beta_{9} + \)\(39\!\cdots\!80\)\( \beta_{8} + \)\(76\!\cdots\!97\)\( \beta_{7} + \)\(11\!\cdots\!66\)\( \beta_{6} + \)\(30\!\cdots\!04\)\( \beta_{5} - \)\(17\!\cdots\!57\)\( \beta_{4} + \)\(20\!\cdots\!75\)\( \beta_{3} - \)\(17\!\cdots\!06\)\( \beta_{2} + \)\(88\!\cdots\!28\)\( \beta_{1} + \)\(13\!\cdots\!60\)\(\)\()/62208\)
\(\nu^{7}\)\(=\)\((\)\(\)\(48\!\cdots\!40\)\( \beta_{9} + \)\(34\!\cdots\!40\)\( \beta_{8} + \)\(23\!\cdots\!23\)\( \beta_{7} + \)\(55\!\cdots\!62\)\( \beta_{6} + \)\(31\!\cdots\!32\)\( \beta_{5} + \)\(80\!\cdots\!97\)\( \beta_{4} + \)\(12\!\cdots\!05\)\( \beta_{3} - \)\(48\!\cdots\!14\)\( \beta_{2} + \)\(24\!\cdots\!60\)\( \beta_{1} + \)\(80\!\cdots\!76\)\(\)\()/124416\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(80\!\cdots\!00\)\( \beta_{9} + \)\(41\!\cdots\!36\)\( \beta_{8} + \)\(94\!\cdots\!93\)\( \beta_{7} + \)\(10\!\cdots\!06\)\( \beta_{6} + \)\(34\!\cdots\!88\)\( \beta_{5} - \)\(16\!\cdots\!89\)\( \beta_{4} + \)\(18\!\cdots\!31\)\( \beta_{3} - \)\(21\!\cdots\!58\)\( \beta_{2} + \)\(95\!\cdots\!16\)\( \beta_{1} + \)\(10\!\cdots\!56\)\(\)\()/124416\)
\(\nu^{9}\)\(=\)\((\)\(\)\(70\!\cdots\!80\)\( \beta_{9} + \)\(49\!\cdots\!16\)\( \beta_{8} + \)\(28\!\cdots\!03\)\( \beta_{7} + \)\(10\!\cdots\!94\)\( \beta_{6} + \)\(46\!\cdots\!16\)\( \beta_{5} + \)\(83\!\cdots\!61\)\( \beta_{4} + \)\(18\!\cdots\!45\)\( \beta_{3} - \)\(60\!\cdots\!14\)\( \beta_{2} + \)\(26\!\cdots\!84\)\( \beta_{1} + \)\(10\!\cdots\!52\)\(\)\()/31104\)

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
7.04177e16
5.22279e16
4.51889e16
2.15657e16
−1.93174e14
−9.26226e15
−3.19227e16
−3.95584e16
−4.85771e16
−5.98866e16
−1.59803e18 −3.81224e28 1.88909e36 5.86174e40 6.09207e46 −2.05200e50 −1.95675e54 8.54313e56 −9.36724e58
1.2 −1.16148e18 2.32093e28 6.84414e35 −4.71757e41 −2.69571e46 −1.44318e49 −2.29977e52 −6.03321e55 5.47935e59
1.3 −9.92540e17 9.16621e27 3.20523e35 6.42021e41 −9.09783e45 1.00631e50 3.41525e53 −5.14984e56 −6.37232e59
1.4 −4.25584e17 −2.69872e28 −4.83493e35 −2.34399e41 1.14853e46 5.11468e49 4.88615e53 1.29303e56 9.97565e58
1.5 9.66294e16 4.03037e28 −6.55277e35 −8.26042e40 3.89453e45 −7.51518e49 −1.27540e53 1.02539e57 −7.98200e57
1.6 3.14287e17 −2.85616e27 −5.65837e35 3.01555e41 −8.97656e44 −2.21350e50 −3.86715e53 −5.90846e56 9.47749e58
1.7 8.58138e17 9.54518e27 7.17860e34 −1.94208e41 8.19107e45 3.78566e50 −5.08728e53 −5.07893e56 −1.66657e59
1.8 1.04139e18 −4.57629e28 4.19888e35 5.48255e41 −4.76572e46 1.41978e50 −2.54857e53 1.49524e57 5.70949e59
1.9 1.25784e18 −1.09661e28 9.17555e35 −6.18260e41 −1.37937e46 −3.09107e50 3.18161e53 −4.78748e56 −7.77674e59
1.10 1.52927e18 3.29001e28 1.67406e36 6.57038e41 5.03133e46 −6.63364e49 1.54372e54 4.83416e56 1.00479e60
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{120}^{\mathrm{new}}(\Gamma_0(1))\).