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: \(59.5852992940\)
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 \( 2^{83}\cdot 3^{30}\cdot 5^{10}\cdot 7^{8}\cdot 11^{2}\cdot 19 \)
Twist minimal: yes
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.

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

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.98.a.a 7
3.b odd 2 1 9.98.a.a 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.98.a.a 7 1.a even 1 1 trivial
9.98.a.a 7 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{98}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 16697241085008 T + \)\(47\!\cdots\!92\)\( T^{2} - \)\(49\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!92\)\( T^{4} - \)\(17\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!04\)\( T^{6} - \)\(16\!\cdots\!20\)\( T^{7} + \)\(43\!\cdots\!88\)\( T^{8} - \)\(43\!\cdots\!56\)\( T^{9} + \)\(50\!\cdots\!16\)\( T^{10} - \)\(31\!\cdots\!40\)\( T^{11} + \)\(47\!\cdots\!44\)\( T^{12} + \)\(26\!\cdots\!32\)\( T^{13} + \)\(25\!\cdots\!88\)\( T^{14} \)
$3$ \( 1 - \)\(10\!\cdots\!96\)\( T + \)\(54\!\cdots\!53\)\( T^{2} - \)\(44\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!77\)\( T^{4} - \)\(11\!\cdots\!72\)\( T^{5} + \)\(40\!\cdots\!61\)\( T^{6} - \)\(23\!\cdots\!20\)\( T^{7} + \)\(77\!\cdots\!43\)\( T^{8} - \)\(40\!\cdots\!68\)\( T^{9} + \)\(12\!\cdots\!19\)\( T^{10} - \)\(59\!\cdots\!40\)\( T^{11} + \)\(13\!\cdots\!79\)\( T^{12} - \)\(48\!\cdots\!64\)\( T^{13} + \)\(92\!\cdots\!67\)\( T^{14} \)
$5$ \( 1 + \)\(36\!\cdots\!50\)\( T + \)\(23\!\cdots\!75\)\( T^{2} + \)\(24\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!25\)\( T^{4} - \)\(35\!\cdots\!50\)\( T^{5} + \)\(17\!\cdots\!75\)\( T^{6} - \)\(48\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!75\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{9} + \)\(63\!\cdots\!25\)\( T^{10} + \)\(39\!\cdots\!00\)\( T^{11} + \)\(23\!\cdots\!75\)\( T^{12} + \)\(23\!\cdots\!50\)\( T^{13} + \)\(39\!\cdots\!25\)\( T^{14} \)
$7$ \( 1 + \)\(18\!\cdots\!08\)\( T + \)\(47\!\cdots\!57\)\( T^{2} + \)\(72\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!97\)\( T^{4} + \)\(13\!\cdots\!76\)\( T^{5} + \)\(15\!\cdots\!29\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!03\)\( T^{8} + \)\(12\!\cdots\!24\)\( T^{9} + \)\(93\!\cdots\!71\)\( T^{10} + \)\(56\!\cdots\!00\)\( T^{11} + \)\(35\!\cdots\!99\)\( T^{12} + \)\(12\!\cdots\!92\)\( T^{13} + \)\(66\!\cdots\!43\)\( T^{14} \)
$11$ \( 1 + \)\(21\!\cdots\!96\)\( T + \)\(24\!\cdots\!61\)\( T^{2} + \)\(84\!\cdots\!16\)\( T^{3} + \)\(47\!\cdots\!41\)\( T^{4} + \)\(15\!\cdots\!48\)\( T^{5} + \)\(70\!\cdots\!33\)\( T^{6} + \)\(17\!\cdots\!28\)\( T^{7} + \)\(73\!\cdots\!43\)\( T^{8} + \)\(16\!\cdots\!68\)\( T^{9} + \)\(52\!\cdots\!51\)\( T^{10} + \)\(97\!\cdots\!96\)\( T^{11} + \)\(29\!\cdots\!11\)\( T^{12} + \)\(26\!\cdots\!16\)\( T^{13} + \)\(12\!\cdots\!91\)\( T^{14} \)
$13$ \( 1 + \)\(88\!\cdots\!14\)\( T + \)\(40\!\cdots\!63\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!57\)\( T^{4} + \)\(44\!\cdots\!38\)\( T^{5} + \)\(11\!\cdots\!51\)\( T^{6} + \)\(57\!\cdots\!60\)\( T^{7} + \)\(13\!\cdots\!83\)\( T^{8} + \)\(57\!\cdots\!82\)\( T^{9} + \)\(11\!\cdots\!09\)\( T^{10} + \)\(43\!\cdots\!20\)\( T^{11} + \)\(73\!\cdots\!59\)\( T^{12} + \)\(18\!\cdots\!66\)\( T^{13} + \)\(23\!\cdots\!77\)\( T^{14} \)
$17$ \( 1 - \)\(16\!\cdots\!42\)\( T + \)\(70\!\cdots\!87\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!57\)\( T^{4} - \)\(55\!\cdots\!54\)\( T^{5} + \)\(90\!\cdots\!79\)\( T^{6} - \)\(15\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!83\)\( T^{8} - \)\(28\!\cdots\!66\)\( T^{9} + \)\(35\!\cdots\!81\)\( T^{10} - \)\(36\!\cdots\!20\)\( T^{11} + \)\(41\!\cdots\!59\)\( T^{12} - \)\(21\!\cdots\!38\)\( T^{13} + \)\(29\!\cdots\!53\)\( T^{14} \)
$19$ \( 1 - \)\(13\!\cdots\!20\)\( T + \)\(52\!\cdots\!73\)\( T^{2} - \)\(49\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} - \)\(81\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!65\)\( T^{6} - \)\(93\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!35\)\( T^{8} - \)\(97\!\cdots\!00\)\( T^{9} + \)\(14\!\cdots\!79\)\( T^{10} - \)\(71\!\cdots\!80\)\( T^{11} + \)\(82\!\cdots\!27\)\( T^{12} - \)\(22\!\cdots\!20\)\( T^{13} + \)\(18\!\cdots\!79\)\( T^{14} \)
$23$ \( 1 - \)\(92\!\cdots\!76\)\( T + \)\(50\!\cdots\!73\)\( T^{2} - \)\(24\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!37\)\( T^{4} - \)\(73\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!41\)\( T^{6} + \)\(21\!\cdots\!40\)\( T^{7} + \)\(14\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!68\)\( T^{9} + \)\(18\!\cdots\!99\)\( T^{10} - \)\(55\!\cdots\!20\)\( T^{11} + \)\(13\!\cdots\!39\)\( T^{12} - \)\(31\!\cdots\!04\)\( T^{13} + \)\(41\!\cdots\!87\)\( T^{14} \)
$29$ \( 1 + \)\(11\!\cdots\!70\)\( T + \)\(34\!\cdots\!63\)\( T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(60\!\cdots\!01\)\( T^{4} + \)\(48\!\cdots\!50\)\( T^{5} + \)\(64\!\cdots\!15\)\( T^{6} + \)\(42\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!35\)\( T^{8} + \)\(24\!\cdots\!50\)\( T^{9} + \)\(21\!\cdots\!29\)\( T^{10} + \)\(86\!\cdots\!80\)\( T^{11} + \)\(64\!\cdots\!87\)\( T^{12} + \)\(15\!\cdots\!70\)\( T^{13} + \)\(92\!\cdots\!69\)\( T^{14} \)
$31$ \( 1 - \)\(17\!\cdots\!84\)\( T + \)\(94\!\cdots\!01\)\( T^{2} - \)\(16\!\cdots\!24\)\( T^{3} + \)\(76\!\cdots\!21\)\( T^{4} - \)\(15\!\cdots\!52\)\( T^{5} + \)\(42\!\cdots\!93\)\( T^{6} - \)\(74\!\cdots\!52\)\( T^{7} + \)\(19\!\cdots\!23\)\( T^{8} - \)\(32\!\cdots\!92\)\( T^{9} + \)\(73\!\cdots\!51\)\( T^{10} - \)\(74\!\cdots\!84\)\( T^{11} + \)\(19\!\cdots\!51\)\( T^{12} - \)\(16\!\cdots\!24\)\( T^{13} + \)\(43\!\cdots\!71\)\( T^{14} \)
$37$ \( 1 + \)\(45\!\cdots\!58\)\( T + \)\(35\!\cdots\!47\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(68\!\cdots\!77\)\( T^{4} + \)\(49\!\cdots\!86\)\( T^{5} + \)\(10\!\cdots\!79\)\( T^{6} + \)\(79\!\cdots\!20\)\( T^{7} + \)\(13\!\cdots\!43\)\( T^{8} + \)\(84\!\cdots\!54\)\( T^{9} + \)\(15\!\cdots\!01\)\( T^{10} + \)\(61\!\cdots\!40\)\( T^{11} + \)\(13\!\cdots\!79\)\( T^{12} + \)\(22\!\cdots\!02\)\( T^{13} + \)\(64\!\cdots\!73\)\( T^{14} \)
$41$ \( 1 - \)\(12\!\cdots\!74\)\( T + \)\(12\!\cdots\!71\)\( T^{2} - \)\(31\!\cdots\!44\)\( T^{3} + \)\(70\!\cdots\!61\)\( T^{4} - \)\(24\!\cdots\!02\)\( T^{5} + \)\(25\!\cdots\!23\)\( T^{6} - \)\(94\!\cdots\!92\)\( T^{7} + \)\(69\!\cdots\!63\)\( T^{8} - \)\(18\!\cdots\!22\)\( T^{9} + \)\(14\!\cdots\!01\)\( T^{10} - \)\(17\!\cdots\!24\)\( T^{11} + \)\(19\!\cdots\!71\)\( T^{12} - \)\(54\!\cdots\!94\)\( T^{13} + \)\(12\!\cdots\!61\)\( T^{14} \)
$43$ \( 1 + \)\(65\!\cdots\!44\)\( T + \)\(67\!\cdots\!93\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!97\)\( T^{4} - \)\(34\!\cdots\!32\)\( T^{5} + \)\(30\!\cdots\!21\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(85\!\cdots\!03\)\( T^{8} - \)\(27\!\cdots\!68\)\( T^{9} + \)\(42\!\cdots\!79\)\( T^{10} - \)\(67\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!99\)\( T^{12} + \)\(31\!\cdots\!56\)\( T^{13} + \)\(13\!\cdots\!07\)\( T^{14} \)
$47$ \( 1 - \)\(15\!\cdots\!92\)\( T + \)\(71\!\cdots\!77\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!37\)\( T^{4} - \)\(38\!\cdots\!44\)\( T^{5} + \)\(63\!\cdots\!29\)\( T^{6} - \)\(76\!\cdots\!40\)\( T^{7} + \)\(99\!\cdots\!23\)\( T^{8} - \)\(94\!\cdots\!36\)\( T^{9} + \)\(10\!\cdots\!11\)\( T^{10} - \)\(69\!\cdots\!80\)\( T^{11} + \)\(66\!\cdots\!39\)\( T^{12} - \)\(22\!\cdots\!28\)\( T^{13} + \)\(22\!\cdots\!83\)\( T^{14} \)
$53$ \( 1 + \)\(65\!\cdots\!54\)\( T + \)\(10\!\cdots\!03\)\( T^{2} + \)\(46\!\cdots\!60\)\( T^{3} + \)\(43\!\cdots\!77\)\( T^{4} + \)\(14\!\cdots\!78\)\( T^{5} + \)\(10\!\cdots\!11\)\( T^{6} + \)\(28\!\cdots\!80\)\( T^{7} + \)\(19\!\cdots\!43\)\( T^{8} + \)\(45\!\cdots\!82\)\( T^{9} + \)\(25\!\cdots\!69\)\( T^{10} + \)\(48\!\cdots\!60\)\( T^{11} + \)\(19\!\cdots\!79\)\( T^{12} + \)\(22\!\cdots\!86\)\( T^{13} + \)\(60\!\cdots\!17\)\( T^{14} \)
$59$ \( 1 + \)\(28\!\cdots\!40\)\( T + \)\(60\!\cdots\!33\)\( T^{2} + \)\(91\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!81\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!65\)\( T^{6} + \)\(90\!\cdots\!00\)\( T^{7} + \)\(65\!\cdots\!35\)\( T^{8} + \)\(42\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!79\)\( T^{10} + \)\(11\!\cdots\!60\)\( T^{11} + \)\(44\!\cdots\!67\)\( T^{12} + \)\(12\!\cdots\!40\)\( T^{13} + \)\(25\!\cdots\!39\)\( T^{14} \)
$61$ \( 1 + \)\(17\!\cdots\!46\)\( T + \)\(20\!\cdots\!11\)\( T^{2} + \)\(16\!\cdots\!16\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} + \)\(60\!\cdots\!98\)\( T^{5} + \)\(28\!\cdots\!83\)\( T^{6} + \)\(11\!\cdots\!28\)\( T^{7} + \)\(43\!\cdots\!43\)\( T^{8} + \)\(13\!\cdots\!18\)\( T^{9} + \)\(37\!\cdots\!01\)\( T^{10} + \)\(86\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!11\)\( T^{12} + \)\(20\!\cdots\!66\)\( T^{13} + \)\(17\!\cdots\!41\)\( T^{14} \)
$67$ \( 1 + \)\(18\!\cdots\!08\)\( T + \)\(22\!\cdots\!37\)\( T^{2} + \)\(19\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!57\)\( T^{4} + \)\(78\!\cdots\!96\)\( T^{5} + \)\(37\!\cdots\!29\)\( T^{6} + \)\(14\!\cdots\!40\)\( T^{7} + \)\(50\!\cdots\!83\)\( T^{8} + \)\(14\!\cdots\!84\)\( T^{9} + \)\(33\!\cdots\!31\)\( T^{10} + \)\(65\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!59\)\( T^{12} + \)\(10\!\cdots\!12\)\( T^{13} + \)\(80\!\cdots\!03\)\( T^{14} \)
$71$ \( 1 + \)\(36\!\cdots\!56\)\( T + \)\(16\!\cdots\!81\)\( T^{2} + \)\(12\!\cdots\!96\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} + \)\(37\!\cdots\!48\)\( T^{5} + \)\(72\!\cdots\!13\)\( T^{6} + \)\(13\!\cdots\!88\)\( T^{7} + \)\(27\!\cdots\!83\)\( T^{8} + \)\(52\!\cdots\!88\)\( T^{9} + \)\(69\!\cdots\!51\)\( T^{10} + \)\(23\!\cdots\!56\)\( T^{11} + \)\(11\!\cdots\!31\)\( T^{12} + \)\(99\!\cdots\!96\)\( T^{13} + \)\(10\!\cdots\!31\)\( T^{14} \)
$73$ \( 1 - \)\(59\!\cdots\!26\)\( T + \)\(46\!\cdots\!23\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(83\!\cdots\!37\)\( T^{4} - \)\(25\!\cdots\!02\)\( T^{5} + \)\(79\!\cdots\!91\)\( T^{6} - \)\(18\!\cdots\!60\)\( T^{7} + \)\(44\!\cdots\!23\)\( T^{8} - \)\(76\!\cdots\!18\)\( T^{9} + \)\(14\!\cdots\!49\)\( T^{10} - \)\(17\!\cdots\!20\)\( T^{11} + \)\(23\!\cdots\!39\)\( T^{12} - \)\(16\!\cdots\!54\)\( T^{13} + \)\(15\!\cdots\!37\)\( T^{14} \)
$79$ \( 1 - \)\(15\!\cdots\!80\)\( T + \)\(80\!\cdots\!13\)\( T^{2} - \)\(99\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!01\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!65\)\( T^{6} - \)\(41\!\cdots\!00\)\( T^{7} + \)\(62\!\cdots\!35\)\( T^{8} - \)\(37\!\cdots\!00\)\( T^{9} + \)\(44\!\cdots\!79\)\( T^{10} - \)\(18\!\cdots\!20\)\( T^{11} + \)\(18\!\cdots\!87\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{13} + \)\(30\!\cdots\!19\)\( T^{14} \)
$83$ \( 1 + \)\(27\!\cdots\!84\)\( T + \)\(55\!\cdots\!33\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!17\)\( T^{4} + \)\(26\!\cdots\!08\)\( T^{5} + \)\(37\!\cdots\!81\)\( T^{6} + \)\(47\!\cdots\!20\)\( T^{7} + \)\(52\!\cdots\!63\)\( T^{8} + \)\(53\!\cdots\!32\)\( T^{9} + \)\(54\!\cdots\!39\)\( T^{10} + \)\(43\!\cdots\!40\)\( T^{11} + \)\(31\!\cdots\!19\)\( T^{12} + \)\(21\!\cdots\!76\)\( T^{13} + \)\(11\!\cdots\!47\)\( T^{14} \)
$89$ \( 1 - \)\(86\!\cdots\!90\)\( T + \)\(19\!\cdots\!03\)\( T^{2} - \)\(78\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} - \)\(84\!\cdots\!50\)\( T^{5} + \)\(32\!\cdots\!15\)\( T^{6} - \)\(74\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!35\)\( T^{8} - \)\(12\!\cdots\!50\)\( T^{9} + \)\(25\!\cdots\!29\)\( T^{10} - \)\(18\!\cdots\!60\)\( T^{11} + \)\(56\!\cdots\!47\)\( T^{12} - \)\(30\!\cdots\!90\)\( T^{13} + \)\(43\!\cdots\!09\)\( T^{14} \)
$97$ \( 1 + \)\(46\!\cdots\!58\)\( T + \)\(19\!\cdots\!27\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!37\)\( T^{4} + \)\(40\!\cdots\!06\)\( T^{5} + \)\(11\!\cdots\!79\)\( T^{6} + \)\(29\!\cdots\!60\)\( T^{7} + \)\(59\!\cdots\!23\)\( T^{8} + \)\(10\!\cdots\!14\)\( T^{9} + \)\(26\!\cdots\!61\)\( T^{10} + \)\(19\!\cdots\!20\)\( T^{11} + \)\(76\!\cdots\!39\)\( T^{12} + \)\(92\!\cdots\!22\)\( T^{13} + \)\(10\!\cdots\!33\)\( T^{14} \)
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