Properties

Label 1.98.a.a
Level 1
Weight 98
Character orbit 1.a
Self dual Yes
Analytic conductor 59.585
Analytic rank 1
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 98 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(59.585299294\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{30}\cdot 5^{10}\cdot 7^{8}\cdot 11^{2}\cdot 19 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(-2385320155001 + \beta_{1}) q^{2}\) \(+(\)\(14\!\cdots\!54\)\( - 23858416 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(23\!\cdots\!64\)\( + 26545654756233 \beta_{1} - 168236 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(52\!\cdots\!91\)\( + 1855342166323926078 \beta_{1} + 7898303201 \beta_{2} - 6690 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(43\!\cdots\!73\)\( + \)\(35\!\cdots\!39\)\( \beta_{1} + 35204970774288 \beta_{2} - 16394794 \beta_{3} + 463 \beta_{4} - \beta_{5}) q^{6}\) \(+(-\)\(26\!\cdots\!39\)\( + \)\(19\!\cdots\!09\)\( \beta_{1} + 30034044083933382 \beta_{2} + 42071231678 \beta_{3} + 2835738 \beta_{4} - 37 \beta_{5} + \beta_{6}) q^{7}\) \(+(\)\(51\!\cdots\!48\)\( - \)\(43\!\cdots\!20\)\( \beta_{1} - 6435319064182925120 \beta_{2} - 7762564472976 \beta_{3} - 554364256 \beta_{4} + 25824 \beta_{5} - 192 \beta_{6}) q^{8}\) \(+(\)\(49\!\cdots\!03\)\( - \)\(54\!\cdots\!52\)\( \beta_{1} - \)\(74\!\cdots\!74\)\( \beta_{2} - 6530297043927348 \beta_{3} - 541723019994 \beta_{4} + 63549828 \beta_{5} - 178740 \beta_{6}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-2385320155001 + \beta_{1}) q^{2}\) \(+(\)\(14\!\cdots\!54\)\( - 23858416 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(23\!\cdots\!64\)\( + 26545654756233 \beta_{1} - 168236 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(52\!\cdots\!91\)\( + 1855342166323926078 \beta_{1} + 7898303201 \beta_{2} - 6690 \beta_{3} + \beta_{4}) q^{5}\) \(+(-\)\(43\!\cdots\!73\)\( + \)\(35\!\cdots\!39\)\( \beta_{1} + 35204970774288 \beta_{2} - 16394794 \beta_{3} + 463 \beta_{4} - \beta_{5}) q^{6}\) \(+(-\)\(26\!\cdots\!39\)\( + \)\(19\!\cdots\!09\)\( \beta_{1} + 30034044083933382 \beta_{2} + 42071231678 \beta_{3} + 2835738 \beta_{4} - 37 \beta_{5} + \beta_{6}) q^{7}\) \(+(\)\(51\!\cdots\!48\)\( - \)\(43\!\cdots\!20\)\( \beta_{1} - 6435319064182925120 \beta_{2} - 7762564472976 \beta_{3} - 554364256 \beta_{4} + 25824 \beta_{5} - 192 \beta_{6}) q^{8}\) \(+(\)\(49\!\cdots\!03\)\( - \)\(54\!\cdots\!52\)\( \beta_{1} - \)\(74\!\cdots\!74\)\( \beta_{2} - 6530297043927348 \beta_{3} - 541723019994 \beta_{4} + 63549828 \beta_{5} - 178740 \beta_{6}) q^{9}\) \(+(\)\(33\!\cdots\!98\)\( - \)\(14\!\cdots\!34\)\( \beta_{1} - \)\(16\!\cdots\!28\)\( \beta_{2} + 449874125925846120 \beta_{3} - 50699164356828 \beta_{4} + 37758784100 \beta_{5} - 6310400 \beta_{6}) q^{10}\) \(+(-\)\(30\!\cdots\!64\)\( - \)\(15\!\cdots\!42\)\( \beta_{1} - \)\(18\!\cdots\!23\)\( \beta_{2} + \)\(22\!\cdots\!80\)\( \beta_{3} - 5133460004335996 \beta_{4} + 4883428564782 \beta_{5} + 2139845610 \beta_{6}) q^{11}\) \(+(\)\(42\!\cdots\!84\)\( - \)\(24\!\cdots\!00\)\( \beta_{1} - \)\(14\!\cdots\!64\)\( \beta_{2} + \)\(20\!\cdots\!36\)\( \beta_{3} - 2156500376186966784 \beta_{4} - 99871873323264 \beta_{5} - 143542015488 \beta_{6}) q^{12}\) \(+(-\)\(12\!\cdots\!11\)\( + \)\(19\!\cdots\!90\)\( \beta_{1} - \)\(37\!\cdots\!27\)\( \beta_{2} + \)\(30\!\cdots\!34\)\( \beta_{3} - \)\(10\!\cdots\!51\)\( \beta_{4} - 9664503066346856 \beta_{5} + 5114833293128 \beta_{6}) q^{13}\) \(+(\)\(35\!\cdots\!50\)\( - \)\(21\!\cdots\!66\)\( \beta_{1} - \)\(13\!\cdots\!48\)\( \beta_{2} - \)\(61\!\cdots\!72\)\( \beta_{3} - \)\(14\!\cdots\!30\)\( \beta_{4} + 440342187813109230 \beta_{5} - 106449797253120 \beta_{6}) q^{14}\) \(+(-\)\(20\!\cdots\!81\)\( + \)\(27\!\cdots\!23\)\( \beta_{1} + \)\(17\!\cdots\!66\)\( \beta_{2} - \)\(52\!\cdots\!90\)\( \beta_{3} + \)\(10\!\cdots\!66\)\( \beta_{4} - 5592888399807242075 \beta_{5} + 882934739618175 \beta_{6}) q^{15}\) \(+(-\)\(44\!\cdots\!32\)\( - \)\(24\!\cdots\!68\)\( \beta_{1} + \)\(32\!\cdots\!72\)\( \beta_{2} - \)\(14\!\cdots\!48\)\( \beta_{3} + \)\(31\!\cdots\!68\)\( \beta_{4} - 82722739840767428096 \beta_{5} + 23133274116930560 \beta_{6}) q^{16}\) \(+(\)\(23\!\cdots\!04\)\( - \)\(28\!\cdots\!76\)\( \beta_{1} + \)\(23\!\cdots\!82\)\( \beta_{2} - \)\(15\!\cdots\!88\)\( \beta_{3} - \)\(42\!\cdots\!98\)\( \beta_{4} + \)\(42\!\cdots\!52\)\( \beta_{5} - 1137697503697956996 \beta_{6}) q^{17}\) \(+(-\)\(10\!\cdots\!85\)\( + \)\(41\!\cdots\!45\)\( \beta_{1} + \)\(12\!\cdots\!68\)\( \beta_{2} + \)\(10\!\cdots\!12\)\( \beta_{3} + \)\(37\!\cdots\!52\)\( \beta_{4} - \)\(73\!\cdots\!48\)\( \beta_{5} + 27253415869653832704 \beta_{6}) q^{18}\) \(+(\)\(19\!\cdots\!56\)\( - \)\(20\!\cdots\!02\)\( \beta_{1} + \)\(20\!\cdots\!75\)\( \beta_{2} + \)\(41\!\cdots\!84\)\( \beta_{3} + \)\(33\!\cdots\!28\)\( \beta_{4} + \)\(61\!\cdots\!74\)\( \beta_{5} - \)\(46\!\cdots\!30\)\( \beta_{6}) q^{19}\) \(+(-\)\(18\!\cdots\!92\)\( + \)\(94\!\cdots\!86\)\( \beta_{1} + \)\(53\!\cdots\!12\)\( \beta_{2} + \)\(13\!\cdots\!70\)\( \beta_{3} - \)\(67\!\cdots\!88\)\( \beta_{4} + \)\(38\!\cdots\!00\)\( \beta_{5} + \)\(61\!\cdots\!00\)\( \beta_{6}) q^{20}\) \(+(-\)\(11\!\cdots\!44\)\( + \)\(25\!\cdots\!16\)\( \beta_{1} + \)\(96\!\cdots\!28\)\( \beta_{2} - \)\(57\!\cdots\!28\)\( \beta_{3} - \)\(17\!\cdots\!60\)\( \beta_{4} - \)\(80\!\cdots\!20\)\( \beta_{5} - \)\(64\!\cdots\!60\)\( \beta_{6}) q^{21}\) \(+(-\)\(27\!\cdots\!59\)\( - \)\(19\!\cdots\!55\)\( \beta_{1} + \)\(84\!\cdots\!72\)\( \beta_{2} - \)\(14\!\cdots\!34\)\( \beta_{3} + \)\(81\!\cdots\!01\)\( \beta_{4} + \)\(11\!\cdots\!81\)\( \beta_{5} + \)\(54\!\cdots\!72\)\( \beta_{6}) q^{22}\) \(+(\)\(13\!\cdots\!95\)\( + \)\(26\!\cdots\!07\)\( \beta_{1} + \)\(22\!\cdots\!26\)\( \beta_{2} + \)\(40\!\cdots\!70\)\( \beta_{3} - \)\(68\!\cdots\!30\)\( \beta_{4} - \)\(77\!\cdots\!55\)\( \beta_{5} - \)\(33\!\cdots\!85\)\( \beta_{6}) q^{23}\) \(+(\)\(24\!\cdots\!52\)\( - \)\(92\!\cdots\!28\)\( \beta_{1} - \)\(18\!\cdots\!88\)\( \beta_{2} + \)\(10\!\cdots\!32\)\( \beta_{3} - \)\(38\!\cdots\!92\)\( \beta_{4} + \)\(16\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!40\)\( \beta_{6}) q^{24}\) \(+(-\)\(21\!\cdots\!25\)\( - \)\(21\!\cdots\!00\)\( \beta_{1} - \)\(25\!\cdots\!00\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3} + \)\(36\!\cdots\!00\)\( \beta_{4} + \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(36\!\cdots\!00\)\( \beta_{6}) q^{25}\) \(+(\)\(36\!\cdots\!02\)\( - \)\(73\!\cdots\!58\)\( \beta_{1} - \)\(15\!\cdots\!76\)\( \beta_{2} - \)\(34\!\cdots\!92\)\( \beta_{3} - \)\(21\!\cdots\!36\)\( \beta_{4} - \)\(32\!\cdots\!48\)\( \beta_{5} - \)\(10\!\cdots\!80\)\( \beta_{6}) q^{26}\) \(+(-\)\(23\!\cdots\!42\)\( - \)\(13\!\cdots\!50\)\( \beta_{1} + \)\(10\!\cdots\!10\)\( \beta_{2} + \)\(97\!\cdots\!84\)\( \beta_{3} - \)\(19\!\cdots\!96\)\( \beta_{4} + \)\(18\!\cdots\!34\)\( \beta_{5} + \)\(10\!\cdots\!78\)\( \beta_{6}) q^{27}\) \(+(\)\(32\!\cdots\!28\)\( - \)\(75\!\cdots\!64\)\( \beta_{1} + \)\(74\!\cdots\!96\)\( \beta_{2} - \)\(87\!\cdots\!84\)\( \beta_{3} + \)\(72\!\cdots\!96\)\( \beta_{4} - \)\(33\!\cdots\!84\)\( \beta_{5} - \)\(65\!\cdots\!28\)\( \beta_{6}) q^{28}\) \(+(-\)\(17\!\cdots\!27\)\( - \)\(86\!\cdots\!50\)\( \beta_{1} + \)\(27\!\cdots\!97\)\( \beta_{2} - \)\(24\!\cdots\!54\)\( \beta_{3} - \)\(19\!\cdots\!59\)\( \beta_{4} - \)\(33\!\cdots\!12\)\( \beta_{5} + \)\(29\!\cdots\!80\)\( \beta_{6}) q^{29}\) \(+(\)\(50\!\cdots\!18\)\( - \)\(89\!\cdots\!94\)\( \beta_{1} - \)\(11\!\cdots\!48\)\( \beta_{2} + \)\(35\!\cdots\!20\)\( \beta_{3} - \)\(15\!\cdots\!98\)\( \beta_{4} + \)\(33\!\cdots\!50\)\( \beta_{5} - \)\(62\!\cdots\!00\)\( \beta_{6}) q^{30}\) \(+(\)\(25\!\cdots\!24\)\( - \)\(38\!\cdots\!76\)\( \beta_{1} - \)\(88\!\cdots\!44\)\( \beta_{2} + \)\(55\!\cdots\!20\)\( \beta_{3} + \)\(11\!\cdots\!72\)\( \beta_{4} - \)\(13\!\cdots\!84\)\( \beta_{5} - \)\(29\!\cdots\!60\)\( \beta_{6}) q^{31}\) \(+(-\)\(80\!\cdots\!40\)\( - \)\(23\!\cdots\!68\)\( \beta_{1} - \)\(12\!\cdots\!76\)\( \beta_{2} + \)\(12\!\cdots\!92\)\( \beta_{3} - \)\(53\!\cdots\!68\)\( \beta_{4} + \)\(50\!\cdots\!32\)\( \beta_{5} + \)\(42\!\cdots\!64\)\( \beta_{6}) q^{32}\) \(+(\)\(45\!\cdots\!22\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(13\!\cdots\!34\)\( \beta_{2} - \)\(13\!\cdots\!92\)\( \beta_{3} - \)\(27\!\cdots\!22\)\( \beta_{4} + \)\(24\!\cdots\!48\)\( \beta_{5} - \)\(28\!\cdots\!64\)\( \beta_{6}) q^{33}\) \(+(-\)\(51\!\cdots\!94\)\( - \)\(17\!\cdots\!78\)\( \beta_{1} + \)\(80\!\cdots\!92\)\( \beta_{2} + \)\(61\!\cdots\!12\)\( \beta_{3} + \)\(13\!\cdots\!28\)\( \beta_{4} - \)\(13\!\cdots\!76\)\( \beta_{5} + \)\(13\!\cdots\!20\)\( \beta_{6}) q^{34}\) \(+(\)\(18\!\cdots\!28\)\( - \)\(45\!\cdots\!24\)\( \beta_{1} - \)\(96\!\cdots\!08\)\( \beta_{2} + \)\(20\!\cdots\!20\)\( \beta_{3} - \)\(82\!\cdots\!08\)\( \beta_{4} + \)\(20\!\cdots\!00\)\( \beta_{5} - \)\(44\!\cdots\!00\)\( \beta_{6}) q^{35}\) \(+(-\)\(30\!\cdots\!12\)\( + \)\(10\!\cdots\!97\)\( \beta_{1} - \)\(12\!\cdots\!68\)\( \beta_{2} - \)\(32\!\cdots\!03\)\( \beta_{3} - \)\(13\!\cdots\!72\)\( \beta_{4} + \)\(14\!\cdots\!44\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6}) q^{36}\) \(+(-\)\(64\!\cdots\!71\)\( + \)\(20\!\cdots\!38\)\( \beta_{1} - \)\(12\!\cdots\!67\)\( \beta_{2} - \)\(36\!\cdots\!22\)\( \beta_{3} + \)\(32\!\cdots\!93\)\( \beta_{4} - \)\(10\!\cdots\!72\)\( \beta_{5} - \)\(16\!\cdots\!24\)\( \beta_{6}) q^{37}\) \(+(-\)\(42\!\cdots\!57\)\( + \)\(67\!\cdots\!31\)\( \beta_{1} + \)\(88\!\cdots\!24\)\( \beta_{2} + \)\(30\!\cdots\!38\)\( \beta_{3} + \)\(13\!\cdots\!63\)\( \beta_{4} + \)\(33\!\cdots\!43\)\( \beta_{5} + \)\(15\!\cdots\!96\)\( \beta_{6}) q^{38}\) \(+(\)\(61\!\cdots\!83\)\( + \)\(20\!\cdots\!51\)\( \beta_{1} + \)\(40\!\cdots\!10\)\( \beta_{2} + \)\(43\!\cdots\!38\)\( \beta_{3} - \)\(74\!\cdots\!34\)\( \beta_{4} - \)\(37\!\cdots\!47\)\( \beta_{5} - \)\(99\!\cdots\!85\)\( \beta_{6}) q^{39}\) \(+(-\)\(36\!\cdots\!80\)\( + \)\(10\!\cdots\!40\)\( \beta_{1} + \)\(20\!\cdots\!80\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} + \)\(18\!\cdots\!80\)\( \beta_{4} - \)\(46\!\cdots\!00\)\( \beta_{5} + \)\(86\!\cdots\!00\)\( \beta_{6}) q^{40}\) \(+(\)\(17\!\cdots\!14\)\( - \)\(16\!\cdots\!76\)\( \beta_{1} - \)\(41\!\cdots\!44\)\( \beta_{2} - \)\(51\!\cdots\!00\)\( \beta_{3} + \)\(68\!\cdots\!32\)\( \beta_{4} - \)\(35\!\cdots\!64\)\( \beta_{5} - \)\(35\!\cdots\!00\)\( \beta_{6}) q^{41}\) \(+(\)\(47\!\cdots\!64\)\( - \)\(10\!\cdots\!68\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} + \)\(60\!\cdots\!28\)\( \beta_{3} - \)\(10\!\cdots\!12\)\( \beta_{4} + \)\(52\!\cdots\!88\)\( \beta_{5} + \)\(43\!\cdots\!76\)\( \beta_{6}) q^{42}\) \(+(-\)\(94\!\cdots\!14\)\( - \)\(12\!\cdots\!00\)\( \beta_{1} + \)\(22\!\cdots\!33\)\( \beta_{2} + \)\(46\!\cdots\!80\)\( \beta_{3} - \)\(56\!\cdots\!80\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(35\!\cdots\!60\)\( \beta_{6}) q^{43}\) \(+(\)\(44\!\cdots\!56\)\( - \)\(40\!\cdots\!28\)\( \beta_{1} + \)\(17\!\cdots\!32\)\( \beta_{2} - \)\(54\!\cdots\!48\)\( \beta_{3} + \)\(18\!\cdots\!88\)\( \beta_{4} + \)\(62\!\cdots\!64\)\( \beta_{5} - \)\(25\!\cdots\!40\)\( \beta_{6}) q^{44}\) \(+(-\)\(36\!\cdots\!63\)\( + \)\(13\!\cdots\!54\)\( \beta_{1} + \)\(23\!\cdots\!93\)\( \beta_{2} - \)\(36\!\cdots\!70\)\( \beta_{3} + \)\(23\!\cdots\!93\)\( \beta_{4} - \)\(56\!\cdots\!00\)\( \beta_{5} + \)\(83\!\cdots\!00\)\( \beta_{6}) q^{45}\) \(+(\)\(47\!\cdots\!54\)\( + \)\(60\!\cdots\!54\)\( \beta_{1} - \)\(13\!\cdots\!64\)\( \beta_{2} + \)\(20\!\cdots\!80\)\( \beta_{3} - \)\(12\!\cdots\!98\)\( \beta_{4} - \)\(12\!\cdots\!54\)\( \beta_{5} - \)\(11\!\cdots\!00\)\( \beta_{6}) q^{46}\) \(+(\)\(22\!\cdots\!86\)\( + \)\(14\!\cdots\!22\)\( \beta_{1} - \)\(45\!\cdots\!76\)\( \beta_{2} + \)\(34\!\cdots\!40\)\( \beta_{3} - \)\(47\!\cdots\!40\)\( \beta_{4} + \)\(30\!\cdots\!50\)\( \beta_{5} - \)\(31\!\cdots\!70\)\( \beta_{6}) q^{47}\) \(+(-\)\(83\!\cdots\!56\)\( + \)\(23\!\cdots\!56\)\( \beta_{1} + \)\(57\!\cdots\!12\)\( \beta_{2} - \)\(18\!\cdots\!12\)\( \beta_{3} + \)\(36\!\cdots\!48\)\( \beta_{4} + \)\(52\!\cdots\!48\)\( \beta_{5} + \)\(22\!\cdots\!96\)\( \beta_{6}) q^{48}\) \(+(\)\(62\!\cdots\!65\)\( - \)\(79\!\cdots\!04\)\( \beta_{1} + \)\(28\!\cdots\!44\)\( \beta_{2} - \)\(22\!\cdots\!40\)\( \beta_{3} - \)\(81\!\cdots\!12\)\( \beta_{4} - \)\(10\!\cdots\!76\)\( \beta_{5} - \)\(59\!\cdots\!00\)\( \beta_{6}) q^{49}\) \(+(-\)\(38\!\cdots\!75\)\( - \)\(26\!\cdots\!25\)\( \beta_{1} + \)\(51\!\cdots\!00\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(58\!\cdots\!00\)\( \beta_{4} - \)\(14\!\cdots\!00\)\( \beta_{5} + \)\(51\!\cdots\!00\)\( \beta_{6}) q^{50}\) \(+(\)\(10\!\cdots\!58\)\( - \)\(10\!\cdots\!42\)\( \beta_{1} - \)\(20\!\cdots\!30\)\( \beta_{2} + \)\(21\!\cdots\!24\)\( \beta_{3} - \)\(13\!\cdots\!52\)\( \beta_{4} + \)\(77\!\cdots\!34\)\( \beta_{5} + \)\(21\!\cdots\!70\)\( \beta_{6}) q^{51}\) \(+(\)\(65\!\cdots\!72\)\( - \)\(76\!\cdots\!70\)\( \beta_{1} - \)\(39\!\cdots\!64\)\( \beta_{2} - \)\(97\!\cdots\!10\)\( \beta_{3} + \)\(12\!\cdots\!60\)\( \beta_{4} - \)\(15\!\cdots\!00\)\( \beta_{5} - \)\(97\!\cdots\!20\)\( \beta_{6}) q^{52}\) \(+(-\)\(93\!\cdots\!79\)\( + \)\(24\!\cdots\!26\)\( \beta_{1} - \)\(22\!\cdots\!63\)\( \beta_{2} - \)\(52\!\cdots\!46\)\( \beta_{3} - \)\(34\!\cdots\!51\)\( \beta_{4} - \)\(41\!\cdots\!96\)\( \beta_{5} + \)\(16\!\cdots\!68\)\( \beta_{6}) q^{53}\) \(+(-\)\(24\!\cdots\!54\)\( + \)\(11\!\cdots\!78\)\( \beta_{1} + \)\(32\!\cdots\!00\)\( \beta_{2} - \)\(17\!\cdots\!56\)\( \beta_{3} + \)\(22\!\cdots\!98\)\( \beta_{4} + \)\(82\!\cdots\!94\)\( \beta_{5} + \)\(22\!\cdots\!60\)\( \beta_{6}) q^{54}\) \(+(-\)\(39\!\cdots\!27\)\( + \)\(36\!\cdots\!41\)\( \beta_{1} + \)\(56\!\cdots\!22\)\( \beta_{2} + \)\(62\!\cdots\!70\)\( \beta_{3} + \)\(27\!\cdots\!22\)\( \beta_{4} - \)\(14\!\cdots\!25\)\( \beta_{5} - \)\(67\!\cdots\!75\)\( \beta_{6}) q^{55}\) \(+(-\)\(19\!\cdots\!68\)\( + \)\(17\!\cdots\!60\)\( \beta_{1} - \)\(52\!\cdots\!12\)\( \beta_{2} + \)\(41\!\cdots\!44\)\( \beta_{3} + \)\(15\!\cdots\!84\)\( \beta_{4} + \)\(11\!\cdots\!52\)\( \beta_{5} + \)\(11\!\cdots\!80\)\( \beta_{6}) q^{56}\) \(+(-\)\(45\!\cdots\!58\)\( - \)\(13\!\cdots\!24\)\( \beta_{1} - \)\(48\!\cdots\!66\)\( \beta_{2} - \)\(29\!\cdots\!80\)\( \beta_{3} - \)\(56\!\cdots\!70\)\( \beta_{4} - \)\(10\!\cdots\!00\)\( \beta_{5} + \)\(19\!\cdots\!40\)\( \beta_{6}) q^{57}\) \(+(-\)\(15\!\cdots\!30\)\( - \)\(56\!\cdots\!78\)\( \beta_{1} - \)\(46\!\cdots\!32\)\( \beta_{2} - \)\(10\!\cdots\!72\)\( \beta_{3} - \)\(92\!\cdots\!92\)\( \beta_{4} + \)\(21\!\cdots\!48\)\( \beta_{5} - \)\(69\!\cdots\!24\)\( \beta_{6}) q^{58}\) \(+(-\)\(41\!\cdots\!26\)\( - \)\(39\!\cdots\!44\)\( \beta_{1} + \)\(17\!\cdots\!73\)\( \beta_{2} + \)\(15\!\cdots\!52\)\( \beta_{3} + \)\(61\!\cdots\!40\)\( \beta_{4} - \)\(50\!\cdots\!20\)\( \beta_{5} - \)\(14\!\cdots\!60\)\( \beta_{6}) q^{59}\) \(+(-\)\(13\!\cdots\!72\)\( + \)\(97\!\cdots\!76\)\( \beta_{1} + \)\(47\!\cdots\!92\)\( \beta_{2} + \)\(12\!\cdots\!20\)\( \beta_{3} - \)\(67\!\cdots\!08\)\( \beta_{4} - \)\(40\!\cdots\!00\)\( \beta_{5} + \)\(49\!\cdots\!00\)\( \beta_{6}) q^{60}\) \(+(-\)\(25\!\cdots\!83\)\( + \)\(21\!\cdots\!10\)\( \beta_{1} - \)\(44\!\cdots\!35\)\( \beta_{2} + \)\(89\!\cdots\!10\)\( \beta_{3} - \)\(25\!\cdots\!75\)\( \beta_{4} + \)\(41\!\cdots\!80\)\( \beta_{5} - \)\(70\!\cdots\!80\)\( \beta_{6}) q^{61}\) \(+(-\)\(69\!\cdots\!08\)\( + \)\(10\!\cdots\!76\)\( \beta_{1} - \)\(14\!\cdots\!04\)\( \beta_{2} - \)\(43\!\cdots\!32\)\( \beta_{3} - \)\(30\!\cdots\!72\)\( \beta_{4} - \)\(69\!\cdots\!72\)\( \beta_{5} - \)\(36\!\cdots\!44\)\( \beta_{6}) q^{62}\) \(+(-\)\(18\!\cdots\!03\)\( + \)\(20\!\cdots\!05\)\( \beta_{1} - \)\(42\!\cdots\!82\)\( \beta_{2} + \)\(36\!\cdots\!98\)\( \beta_{3} + \)\(15\!\cdots\!38\)\( \beta_{4} - \)\(10\!\cdots\!77\)\( \beta_{5} + \)\(10\!\cdots\!41\)\( \beta_{6}) q^{63}\) \(+(-\)\(34\!\cdots\!08\)\( - \)\(10\!\cdots\!72\)\( \beta_{1} + \)\(64\!\cdots\!16\)\( \beta_{2} - \)\(26\!\cdots\!68\)\( \beta_{3} - \)\(80\!\cdots\!04\)\( \beta_{4} + \)\(13\!\cdots\!08\)\( \beta_{5} - \)\(58\!\cdots\!00\)\( \beta_{6}) q^{64}\) \(+(-\)\(58\!\cdots\!16\)\( - \)\(83\!\cdots\!72\)\( \beta_{1} + \)\(12\!\cdots\!76\)\( \beta_{2} + \)\(38\!\cdots\!60\)\( \beta_{3} - \)\(58\!\cdots\!24\)\( \beta_{4} + \)\(11\!\cdots\!00\)\( \beta_{5} - \)\(36\!\cdots\!00\)\( \beta_{6}) q^{65}\) \(+(-\)\(20\!\cdots\!48\)\( - \)\(18\!\cdots\!76\)\( \beta_{1} + \)\(58\!\cdots\!40\)\( \beta_{2} - \)\(75\!\cdots\!88\)\( \beta_{3} + \)\(10\!\cdots\!84\)\( \beta_{4} - \)\(36\!\cdots\!28\)\( \beta_{5} + \)\(10\!\cdots\!60\)\( \beta_{6}) q^{66}\) \(+(-\)\(26\!\cdots\!36\)\( - \)\(20\!\cdots\!46\)\( \beta_{1} - \)\(20\!\cdots\!73\)\( \beta_{2} + \)\(34\!\cdots\!52\)\( \beta_{3} + \)\(91\!\cdots\!52\)\( \beta_{4} - \)\(20\!\cdots\!78\)\( \beta_{5} - \)\(57\!\cdots\!66\)\( \beta_{6}) q^{67}\) \(+(-\)\(35\!\cdots\!00\)\( - \)\(21\!\cdots\!58\)\( \beta_{1} - \)\(23\!\cdots\!40\)\( \beta_{2} - \)\(10\!\cdots\!54\)\( \beta_{3} + \)\(12\!\cdots\!76\)\( \beta_{4} + \)\(78\!\cdots\!96\)\( \beta_{5} - \)\(21\!\cdots\!68\)\( \beta_{6}) q^{68}\) \(+(-\)\(53\!\cdots\!24\)\( + \)\(16\!\cdots\!36\)\( \beta_{1} - \)\(37\!\cdots\!08\)\( \beta_{2} + \)\(63\!\cdots\!04\)\( \beta_{3} - \)\(10\!\cdots\!08\)\( \beta_{4} - \)\(53\!\cdots\!24\)\( \beta_{5} + \)\(39\!\cdots\!40\)\( \beta_{6}) q^{69}\) \(+(-\)\(83\!\cdots\!84\)\( + \)\(45\!\cdots\!72\)\( \beta_{1} + \)\(44\!\cdots\!24\)\( \beta_{2} - \)\(11\!\cdots\!60\)\( \beta_{3} + \)\(91\!\cdots\!24\)\( \beta_{4} - \)\(19\!\cdots\!00\)\( \beta_{5} + \)\(28\!\cdots\!00\)\( \beta_{6}) q^{70}\) \(+(-\)\(52\!\cdots\!23\)\( + \)\(29\!\cdots\!05\)\( \beta_{1} + \)\(23\!\cdots\!70\)\( \beta_{2} - \)\(15\!\cdots\!70\)\( \beta_{3} + \)\(21\!\cdots\!50\)\( \beta_{4} + \)\(36\!\cdots\!15\)\( \beta_{5} - \)\(11\!\cdots\!15\)\( \beta_{6}) q^{71}\) \(+(\)\(35\!\cdots\!16\)\( - \)\(42\!\cdots\!08\)\( \beta_{1} + \)\(73\!\cdots\!68\)\( \beta_{2} - \)\(65\!\cdots\!84\)\( \beta_{3} - \)\(29\!\cdots\!64\)\( \beta_{4} + \)\(40\!\cdots\!36\)\( \beta_{5} - \)\(20\!\cdots\!28\)\( \beta_{6}) q^{72}\) \(+(\)\(84\!\cdots\!04\)\( - \)\(23\!\cdots\!04\)\( \beta_{1} - \)\(20\!\cdots\!78\)\( \beta_{2} + \)\(43\!\cdots\!88\)\( \beta_{3} + \)\(22\!\cdots\!38\)\( \beta_{4} + \)\(89\!\cdots\!68\)\( \beta_{5} + \)\(10\!\cdots\!96\)\( \beta_{6}) q^{73}\) \(+(\)\(36\!\cdots\!86\)\( - \)\(44\!\cdots\!66\)\( \beta_{1} - \)\(20\!\cdots\!24\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3} - \)\(19\!\cdots\!28\)\( \beta_{4} - \)\(30\!\cdots\!64\)\( \beta_{5} - \)\(95\!\cdots\!80\)\( \beta_{6}) q^{74}\) \(+(\)\(62\!\cdots\!50\)\( - \)\(63\!\cdots\!00\)\( \beta_{1} + \)\(72\!\cdots\!25\)\( \beta_{2} - \)\(26\!\cdots\!00\)\( \beta_{3} + \)\(55\!\cdots\!00\)\( \beta_{4} + \)\(43\!\cdots\!00\)\( \beta_{5} - \)\(31\!\cdots\!00\)\( \beta_{6}) q^{75}\) \(+(\)\(92\!\cdots\!84\)\( + \)\(37\!\cdots\!40\)\( \beta_{1} + \)\(68\!\cdots\!96\)\( \beta_{2} + \)\(71\!\cdots\!08\)\( \beta_{3} + \)\(27\!\cdots\!48\)\( \beta_{4} + \)\(91\!\cdots\!44\)\( \beta_{5} + \)\(10\!\cdots\!60\)\( \beta_{6}) q^{76}\) \(+(\)\(17\!\cdots\!40\)\( + \)\(39\!\cdots\!16\)\( \beta_{1} + \)\(15\!\cdots\!56\)\( \beta_{2} - \)\(95\!\cdots\!68\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(30\!\cdots\!28\)\( \beta_{5} - \)\(10\!\cdots\!56\)\( \beta_{6}) q^{77}\) \(+(\)\(36\!\cdots\!70\)\( + \)\(58\!\cdots\!78\)\( \beta_{1} - \)\(10\!\cdots\!72\)\( \beta_{2} + \)\(18\!\cdots\!96\)\( \beta_{3} - \)\(76\!\cdots\!54\)\( \beta_{4} + \)\(17\!\cdots\!06\)\( \beta_{5} - \)\(73\!\cdots\!68\)\( \beta_{6}) q^{78}\) \(+(\)\(22\!\cdots\!90\)\( + \)\(63\!\cdots\!14\)\( \beta_{1} + \)\(13\!\cdots\!08\)\( \beta_{2} + \)\(12\!\cdots\!96\)\( \beta_{3} + \)\(23\!\cdots\!08\)\( \beta_{4} + \)\(16\!\cdots\!74\)\( \beta_{5} + \)\(46\!\cdots\!10\)\( \beta_{6}) q^{79}\) \(+(\)\(49\!\cdots\!64\)\( - \)\(10\!\cdots\!12\)\( \beta_{1} - \)\(94\!\cdots\!04\)\( \beta_{2} - \)\(18\!\cdots\!40\)\( \beta_{3} + \)\(88\!\cdots\!96\)\( \beta_{4} + \)\(33\!\cdots\!00\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6}) q^{80}\) \(+(-\)\(11\!\cdots\!57\)\( - \)\(36\!\cdots\!52\)\( \beta_{1} + \)\(67\!\cdots\!94\)\( \beta_{2} - \)\(39\!\cdots\!84\)\( \beta_{3} - \)\(18\!\cdots\!10\)\( \beta_{4} - \)\(40\!\cdots\!40\)\( \beta_{5} + \)\(15\!\cdots\!60\)\( \beta_{6}) q^{81}\) \(+(-\)\(29\!\cdots\!38\)\( - \)\(58\!\cdots\!34\)\( \beta_{1} - \)\(22\!\cdots\!24\)\( \beta_{2} - \)\(13\!\cdots\!12\)\( \beta_{3} - \)\(18\!\cdots\!72\)\( \beta_{4} - \)\(46\!\cdots\!12\)\( \beta_{5} + \)\(21\!\cdots\!96\)\( \beta_{6}) q^{82}\) \(+(-\)\(39\!\cdots\!82\)\( - \)\(82\!\cdots\!44\)\( \beta_{1} + \)\(56\!\cdots\!27\)\( \beta_{2} + \)\(37\!\cdots\!00\)\( \beta_{3} - \)\(17\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} - \)\(79\!\cdots\!00\)\( \beta_{6}) q^{83}\) \(+(-\)\(17\!\cdots\!48\)\( + \)\(87\!\cdots\!00\)\( \beta_{1} - \)\(42\!\cdots\!68\)\( \beta_{2} + \)\(12\!\cdots\!36\)\( \beta_{3} + \)\(11\!\cdots\!16\)\( \beta_{4} + \)\(34\!\cdots\!68\)\( \beta_{5} + \)\(17\!\cdots\!00\)\( \beta_{6}) q^{84}\) \(+(-\)\(23\!\cdots\!62\)\( + \)\(48\!\cdots\!96\)\( \beta_{1} + \)\(54\!\cdots\!82\)\( \beta_{2} + \)\(47\!\cdots\!20\)\( \beta_{3} + \)\(41\!\cdots\!82\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5} - \)\(48\!\cdots\!00\)\( \beta_{6}) q^{85}\) \(+(-\)\(22\!\cdots\!19\)\( + \)\(40\!\cdots\!29\)\( \beta_{1} - \)\(39\!\cdots\!56\)\( \beta_{2} - \)\(24\!\cdots\!26\)\( \beta_{3} - \)\(58\!\cdots\!99\)\( \beta_{4} - \)\(11\!\cdots\!47\)\( \beta_{5} - \)\(43\!\cdots\!80\)\( \beta_{6}) q^{86}\) \(+(-\)\(64\!\cdots\!73\)\( + \)\(60\!\cdots\!03\)\( \beta_{1} + \)\(29\!\cdots\!22\)\( \beta_{2} + \)\(47\!\cdots\!58\)\( \beta_{3} + \)\(49\!\cdots\!78\)\( \beta_{4} + \)\(78\!\cdots\!73\)\( \beta_{5} + \)\(68\!\cdots\!11\)\( \beta_{6}) q^{87}\) \(+(-\)\(29\!\cdots\!64\)\( - \)\(71\!\cdots\!40\)\( \beta_{1} - \)\(30\!\cdots\!40\)\( \beta_{2} + \)\(19\!\cdots\!48\)\( \beta_{3} + \)\(20\!\cdots\!08\)\( \beta_{4} + \)\(58\!\cdots\!08\)\( \beta_{5} + \)\(18\!\cdots\!16\)\( \beta_{6}) q^{88}\) \(+(\)\(12\!\cdots\!84\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(20\!\cdots\!86\)\( \beta_{2} + \)\(42\!\cdots\!88\)\( \beta_{3} + \)\(11\!\cdots\!38\)\( \beta_{4} - \)\(41\!\cdots\!96\)\( \beta_{5} - \)\(10\!\cdots\!80\)\( \beta_{6}) q^{89}\) \(+(\)\(24\!\cdots\!14\)\( - \)\(83\!\cdots\!62\)\( \beta_{1} - \)\(12\!\cdots\!04\)\( \beta_{2} + \)\(61\!\cdots\!60\)\( \beta_{3} - \)\(21\!\cdots\!04\)\( \beta_{4} + \)\(42\!\cdots\!00\)\( \beta_{5} - \)\(86\!\cdots\!00\)\( \beta_{6}) q^{90}\) \(+(\)\(33\!\cdots\!04\)\( - \)\(80\!\cdots\!00\)\( \beta_{1} + \)\(67\!\cdots\!28\)\( \beta_{2} - \)\(30\!\cdots\!16\)\( \beta_{3} - \)\(45\!\cdots\!56\)\( \beta_{4} + \)\(21\!\cdots\!32\)\( \beta_{5} + \)\(31\!\cdots\!80\)\( \beta_{6}) q^{91}\) \(+(\)\(88\!\cdots\!92\)\( + \)\(78\!\cdots\!12\)\( \beta_{1} - \)\(61\!\cdots\!36\)\( \beta_{2} + \)\(21\!\cdots\!48\)\( \beta_{3} + \)\(10\!\cdots\!88\)\( \beta_{4} - \)\(10\!\cdots\!52\)\( \beta_{5} + \)\(48\!\cdots\!16\)\( \beta_{6}) q^{92}\) \(+(\)\(23\!\cdots\!20\)\( + \)\(11\!\cdots\!80\)\( \beta_{1} - \)\(23\!\cdots\!04\)\( \beta_{2} - \)\(28\!\cdots\!16\)\( \beta_{3} - \)\(42\!\cdots\!16\)\( \beta_{4} + \)\(29\!\cdots\!24\)\( \beta_{5} - \)\(19\!\cdots\!72\)\( \beta_{6}) q^{93}\) \(+(\)\(25\!\cdots\!60\)\( + \)\(80\!\cdots\!88\)\( \beta_{1} + \)\(44\!\cdots\!40\)\( \beta_{2} + \)\(17\!\cdots\!44\)\( \beta_{3} + \)\(12\!\cdots\!28\)\( \beta_{4} - \)\(59\!\cdots\!16\)\( \beta_{5} + \)\(35\!\cdots\!60\)\( \beta_{6}) q^{94}\) \(+(\)\(15\!\cdots\!15\)\( + \)\(60\!\cdots\!55\)\( \beta_{1} + \)\(17\!\cdots\!10\)\( \beta_{2} - \)\(54\!\cdots\!50\)\( \beta_{3} - \)\(28\!\cdots\!90\)\( \beta_{4} + \)\(45\!\cdots\!25\)\( \beta_{5} + \)\(52\!\cdots\!75\)\( \beta_{6}) q^{95}\) \(+(\)\(39\!\cdots\!48\)\( - \)\(97\!\cdots\!64\)\( \beta_{1} + \)\(28\!\cdots\!36\)\( \beta_{2} - \)\(13\!\cdots\!04\)\( \beta_{3} - \)\(34\!\cdots\!76\)\( \beta_{4} + \)\(21\!\cdots\!52\)\( \beta_{5} - \)\(51\!\cdots\!00\)\( \beta_{6}) q^{96}\) \(+(-\)\(66\!\cdots\!72\)\( - \)\(92\!\cdots\!76\)\( \beta_{1} - \)\(32\!\cdots\!98\)\( \beta_{2} - \)\(50\!\cdots\!96\)\( \beta_{3} + \)\(66\!\cdots\!74\)\( \beta_{4} - \)\(27\!\cdots\!96\)\( \beta_{5} - \)\(78\!\cdots\!32\)\( \beta_{6}) q^{97}\) \(+(-\)\(14\!\cdots\!81\)\( - \)\(30\!\cdots\!87\)\( \beta_{1} - \)\(30\!\cdots\!16\)\( \beta_{2} - \)\(20\!\cdots\!68\)\( \beta_{3} + \)\(87\!\cdots\!92\)\( \beta_{4} - \)\(21\!\cdots\!68\)\( \beta_{5} + \)\(13\!\cdots\!44\)\( \beta_{6}) q^{98}\) \(+(-\)\(19\!\cdots\!22\)\( - \)\(60\!\cdots\!00\)\( \beta_{1} - \)\(30\!\cdots\!67\)\( \beta_{2} + \)\(21\!\cdots\!24\)\( \beta_{3} + \)\(14\!\cdots\!84\)\( \beta_{4} + \)\(56\!\cdots\!52\)\( \beta_{5} + \)\(13\!\cdots\!80\)\( \beta_{6}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut -\mathstrut 16697241085008q^{2} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!96\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!84\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!36\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!08\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!51\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut -\mathstrut 16697241085008q^{2} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!96\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!84\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!36\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!08\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!40\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!51\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!96\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!32\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!14\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!68\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!28\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!42\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!24\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!20\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!96\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!76\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!76\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!60\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!75\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!04\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!60\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!64\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!70\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!84\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!28\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!12\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!72\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!88\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!58\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!40\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!12\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!74\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!44\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!44\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!48\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!50\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!44\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!92\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!24\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!99\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!84\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!12\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!54\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!80\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!80\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!20\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!40\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!46\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!96\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!24\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!76\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!92\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!08\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!36\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!68\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!56\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!20\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!26\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!48\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!40\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!24\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!52\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!80\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!53\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!56\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!84\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!52\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!76\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!80\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!20\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!90\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!44\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!92\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!52\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!08\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!24\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!58\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!56\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!28\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(x^{6}\mathstrut -\mathstrut \) \(276087571804405990752665088\) \(x^{5}\mathstrut -\mathstrut \) \(120083887007184048105302098667336699968\) \(x^{4}\mathstrut +\mathstrut \) \(19373155893793813105741918625198082602849023860244480\) \(x^{3}\mathstrut +\mathstrut \) \(17598423525325549028964286424215233146805218189713573416303513600\) \(x^{2}\mathstrut -\mathstrut \) \(58411389055962077878580852277509525104521481793026738328029906710656876544000\) \(x\mathstrut -\mathstrut \) \(6001611580367435872109665775701806052295874267155389813520177925124906158334928977920000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 7 \)
\(\beta_{2}\)\(=\)\((\)\(39\!\cdots\!11\) \(\nu^{6}\mathstrut -\mathstrut \) \(15\!\cdots\!79\) \(\nu^{5}\mathstrut -\mathstrut \) \(11\!\cdots\!28\) \(\nu^{4}\mathstrut +\mathstrut \) \(22\!\cdots\!24\) \(\nu^{3}\mathstrut +\mathstrut \) \(80\!\cdots\!84\) \(\nu^{2}\mathstrut -\mathstrut \) \(93\!\cdots\!60\) \(\nu\mathstrut -\mathstrut \) \(13\!\cdots\!80\)\()/\)\(10\!\cdots\!76\)
\(\beta_{3}\)\(=\)\((\)\(16\!\cdots\!49\) \(\nu^{6}\mathstrut -\mathstrut \) \(65\!\cdots\!61\) \(\nu^{5}\mathstrut -\mathstrut \) \(47\!\cdots\!52\) \(\nu^{4}\mathstrut +\mathstrut \) \(96\!\cdots\!16\) \(\nu^{3}\mathstrut +\mathstrut \) \(40\!\cdots\!32\) \(\nu^{2}\mathstrut -\mathstrut \) \(78\!\cdots\!28\) \(\nu\mathstrut -\mathstrut \) \(53\!\cdots\!24\)\()/\)\(26\!\cdots\!44\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(92\!\cdots\!71\) \(\nu^{6}\mathstrut -\mathstrut \) \(96\!\cdots\!89\) \(\nu^{5}\mathstrut +\mathstrut \) \(25\!\cdots\!08\) \(\nu^{4}\mathstrut +\mathstrut \) \(26\!\cdots\!08\) \(\nu^{3}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(18\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(35\!\cdots\!00\)\()/\)\(26\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(20\!\cdots\!73\) \(\nu^{6}\mathstrut -\mathstrut \) \(63\!\cdots\!07\) \(\nu^{5}\mathstrut +\mathstrut \) \(36\!\cdots\!04\) \(\nu^{4}\mathstrut +\mathstrut \) \(90\!\cdots\!04\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(42\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(51\!\cdots\!00\)\()/\)\(26\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(28\!\cdots\!71\) \(\nu^{6}\mathstrut +\mathstrut \) \(49\!\cdots\!89\) \(\nu^{5}\mathstrut -\mathstrut \) \(81\!\cdots\!08\) \(\nu^{4}\mathstrut -\mathstrut \) \(33\!\cdots\!08\) \(\nu^{3}\mathstrut +\mathstrut \) \(57\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(37\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(11\!\cdots\!00\)\()/\)\(37\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(7\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(168236\) \(\beta_{2}\mathstrut +\mathstrut \) \(31316295066249\) \(\beta_{1}\mathstrut +\mathstrut \) \(181744504410671833814939351084\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(1614\) \(\beta_{5}\mathstrut -\mathstrut \) \(34647766\) \(\beta_{4}\mathstrut -\mathstrut \) \(37912750497\) \(\beta_{3}\mathstrut -\mathstrut \) \(477450576811043924\) \(\beta_{2}\mathstrut +\mathstrut \) \(19547555433985889802675272513\) \(\beta_{1}\mathstrut +\mathstrut \) \(355722782998101845850844787368382326399018\)\()/6912\)
\(\nu^{4}\)\(=\)\((\)\(83208391554236\) \(\beta_{6}\mathstrut -\mathstrut \) \(322173225820452038\) \(\beta_{5}\mathstrut +\mathstrut \) \(1203064960652939547246\) \(\beta_{4}\mathstrut +\mathstrut \) \(1283733327371338242533046845\) \(\beta_{3}\mathstrut -\mathstrut \) \(185516955121872855506308496725628\) \(\beta_{2}\mathstrut +\mathstrut \) \(60849044235216889713987489361775554704083\) \(\beta_{1}\mathstrut +\mathstrut \) \(222041298424577286469315152646798609699527419879905738510\)\()/20736\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(2135379277243516459866208108\) \(\beta_{6}\mathstrut +\mathstrut \) \(553756769500719929207084875326\) \(\beta_{5}\mathstrut -\mathstrut \) \(10875023680309249523069217515717894\) \(\beta_{4}\mathstrut +\mathstrut \) \(6557193051433625412362411344394717311\) \(\beta_{3}\mathstrut -\mathstrut \) \(477749382858274259514802814627530422435762644\) \(\beta_{2}\mathstrut +\mathstrut \) \(2709781935637313123400665014559090609245611617113480753\) \(\beta_{1}\mathstrut +\mathstrut \) \(76798467983953557158106291460804736587776692031603561605676984958938\)\()/6912\)
\(\nu^{6}\)\(=\)\((\)\(6422606865411460644946957470502343841172\) \(\beta_{6}\mathstrut -\mathstrut \) \(29470703578181510862497172760005739756271170\) \(\beta_{5}\mathstrut +\mathstrut \) \(91581615863698016957272927265465489697697934714\) \(\beta_{4}\mathstrut +\mathstrut \) \(60523756277364538058808149617144370834087158827838015\) \(\beta_{3}\mathstrut -\mathstrut \) \(7047784403296848252487619319492623019100282559413515362004\) \(\beta_{2}\mathstrut +\mathstrut \) \(3603931329614285373201232387251440897369567282245626277370936307825\) \(\beta_{1}\mathstrut +\mathstrut \) \(10260166144931885623461997696599046679936698010473561913603545653456016348451707482\)\()/6912\)

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
−1.21138e13
−1.07106e13
−2.29850e12
−1.00063e11
1.42059e12
1.16497e13
1.21527e13
−5.83847e14 −6.70202e22 1.82421e29 3.35065e33 3.91296e37 6.90378e40 −1.39919e43 −1.45964e46 −1.95627e48
1.2 −5.16494e14 2.18475e23 1.08310e29 −1.00097e34 −1.12841e38 −1.23831e41 2.59003e43 2.86433e46 5.16994e48
1.3 −1.12713e14 −2.23668e23 −1.45752e29 −2.02292e33 2.52104e37 −1.81030e41 3.42884e43 3.09393e46 2.28010e47
1.4 −7.18835e12 1.45423e23 −1.58405e29 1.38920e34 −1.04535e36 −1.18505e40 2.27771e42 2.05969e45 −9.98605e46
1.5 6.58029e13 3.71452e20 −1.54126e29 −8.33150e33 2.44426e34 1.07181e41 −2.05689e43 −1.90879e46 −5.48237e47
1.6 5.56799e14 −1.35917e23 1.51568e29 4.66064e33 −7.56782e37 2.73237e40 −3.83521e42 −6.14732e44 2.59504e48
1.7 5.80945e14 1.62851e23 1.79040e29 −5.19758e33 9.46076e37 −7.14729e40 1.19582e43 7.43249e45 −3.01951e48
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{98}^{\mathrm{new}}(\Gamma_0(1))\).