Properties

Label 1.90.a.a
Level 1
Weight 90
Character orbit 1.a
Self dual Yes
Analytic conductor 50.162
Analytic rank 1
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 90 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(50.1624291928\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{29}\cdot 5^{9}\cdot 7^{5}\cdot 11^{2}\cdot 13^{2} \)
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.

\(f(q)\) \(=\) \( q\) \(+(-4486761478773 - \beta_{1}) q^{2}\) \(+(-\)\(19\!\cdots\!20\)\( + 8262741 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(32\!\cdots\!37\)\( + 4221830345564 \beta_{1} + 31239 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(14\!\cdots\!79\)\( + 45551948905377184 \beta_{1} + 172707388 \beta_{2} + 2871 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(67\!\cdots\!77\)\( + \)\(31\!\cdots\!78\)\( \beta_{1} + 10484168480098 \beta_{2} + 16310470 \beta_{3} - 480 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(55\!\cdots\!87\)\( + \)\(48\!\cdots\!03\)\( \beta_{1} + 4315035800036120 \beta_{2} + 2995501711 \beta_{3} - 86411 \beta_{4} - 59 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(25\!\cdots\!56\)\( - \)\(38\!\cdots\!84\)\( \beta_{1} - 4715431305101366416 \beta_{2} - 4899941440272 \beta_{3} + 164171872 \beta_{4} - 133152 \beta_{5} - 672 \beta_{6}) q^{8}\) \(+(\)\(80\!\cdots\!75\)\( + \)\(18\!\cdots\!96\)\( \beta_{1} - \)\(29\!\cdots\!84\)\( \beta_{2} - 857361085801770 \beta_{3} + 57915115470 \beta_{4} - 65669988 \beta_{5} + 81900 \beta_{6}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-4486761478773 - \beta_{1}) q^{2}\) \(+(-\)\(19\!\cdots\!20\)\( + 8262741 \beta_{1} - \beta_{2}) q^{3}\) \(+(\)\(32\!\cdots\!37\)\( + 4221830345564 \beta_{1} + 31239 \beta_{2} + \beta_{3}) q^{4}\) \(+(\)\(14\!\cdots\!79\)\( + 45551948905377184 \beta_{1} + 172707388 \beta_{2} + 2871 \beta_{3} - \beta_{4}) q^{5}\) \(+(-\)\(67\!\cdots\!77\)\( + \)\(31\!\cdots\!78\)\( \beta_{1} + 10484168480098 \beta_{2} + 16310470 \beta_{3} - 480 \beta_{4} + \beta_{5}) q^{6}\) \(+(\)\(55\!\cdots\!87\)\( + \)\(48\!\cdots\!03\)\( \beta_{1} + 4315035800036120 \beta_{2} + 2995501711 \beta_{3} - 86411 \beta_{4} - 59 \beta_{5} + \beta_{6}) q^{7}\) \(+(-\)\(25\!\cdots\!56\)\( - \)\(38\!\cdots\!84\)\( \beta_{1} - 4715431305101366416 \beta_{2} - 4899941440272 \beta_{3} + 164171872 \beta_{4} - 133152 \beta_{5} - 672 \beta_{6}) q^{8}\) \(+(\)\(80\!\cdots\!75\)\( + \)\(18\!\cdots\!96\)\( \beta_{1} - \)\(29\!\cdots\!84\)\( \beta_{2} - 857361085801770 \beta_{3} + 57915115470 \beta_{4} - 65669988 \beta_{5} + 81900 \beta_{6}) q^{9}\) \(+(-\)\(48\!\cdots\!82\)\( - \)\(34\!\cdots\!22\)\( \beta_{1} - \)\(31\!\cdots\!04\)\( \beta_{2} - 78659792801551768 \beta_{3} + 15974357780608 \beta_{4} - 361573700 \beta_{5} - 5062400 \beta_{6}) q^{10}\) \(+(-\)\(47\!\cdots\!50\)\( + \)\(11\!\cdots\!17\)\( \beta_{1} + \)\(39\!\cdots\!57\)\( \beta_{2} - 2499715634427394218 \beta_{3} + 1431976763236690 \beta_{4} + 318384886578 \beta_{5} + 196384650 \beta_{6}) q^{11}\) \(+(-\)\(14\!\cdots\!16\)\( - \)\(10\!\cdots\!56\)\( \beta_{1} - \)\(20\!\cdots\!28\)\( \beta_{2} - \)\(95\!\cdots\!88\)\( \beta_{3} + 12936913395717888 \beta_{4} - 15657642724608 \beta_{5} - 5068562688 \beta_{6}) q^{12}\) \(+(-\)\(14\!\cdots\!17\)\( + \)\(31\!\cdots\!36\)\( \beta_{1} + \)\(29\!\cdots\!92\)\( \beta_{2} - \)\(37\!\cdots\!97\)\( \beta_{3} - 2257335939581955553 \beta_{4} + 335131492171688 \beta_{5} + 80246616968 \beta_{6}) q^{13}\) \(+(-\)\(47\!\cdots\!86\)\( - \)\(79\!\cdots\!28\)\( \beta_{1} - \)\(38\!\cdots\!88\)\( \beta_{2} - \)\(15\!\cdots\!12\)\( \beta_{3} + 26132472328520664640 \beta_{4} - 1594694108193870 \beta_{5} - 228630604800 \beta_{6}) q^{14}\) \(+(-\)\(56\!\cdots\!31\)\( - \)\(10\!\cdots\!51\)\( \beta_{1} + \)\(51\!\cdots\!68\)\( \beta_{2} - \)\(56\!\cdots\!19\)\( \beta_{3} + \)\(66\!\cdots\!39\)\( \beta_{4} - 110756898795290725 \beta_{5} - 30566516385825 \beta_{6}) q^{15}\) \(+(\)\(16\!\cdots\!12\)\( + \)\(37\!\cdots\!52\)\( \beta_{1} + \)\(15\!\cdots\!52\)\( \beta_{2} + \)\(55\!\cdots\!36\)\( \beta_{3} - \)\(20\!\cdots\!00\)\( \beta_{4} + 3785918355513714176 \beta_{5} + 1148090844454400 \beta_{6}) q^{16}\) \(+(\)\(11\!\cdots\!76\)\( - \)\(53\!\cdots\!36\)\( \beta_{1} + \)\(15\!\cdots\!24\)\( \beta_{2} + \)\(20\!\cdots\!14\)\( \beta_{3} + \)\(17\!\cdots\!86\)\( \beta_{4} - 68882505988460694516 \beta_{5} - 26652581864159076 \beta_{6}) q^{17}\) \(+(-\)\(21\!\cdots\!69\)\( - \)\(18\!\cdots\!57\)\( \beta_{1} + \)\(48\!\cdots\!20\)\( \beta_{2} - \)\(61\!\cdots\!56\)\( \beta_{3} + \)\(78\!\cdots\!56\)\( \beta_{4} + \)\(83\!\cdots\!64\)\( \beta_{5} + 475956533780872704 \beta_{6}) q^{18}\) \(+(\)\(80\!\cdots\!26\)\( - \)\(74\!\cdots\!05\)\( \beta_{1} + \)\(24\!\cdots\!75\)\( \beta_{2} - \)\(18\!\cdots\!82\)\( \beta_{3} - \)\(29\!\cdots\!70\)\( \beta_{4} - \)\(67\!\cdots\!34\)\( \beta_{5} - 7001867487502589450 \beta_{6}) q^{19}\) \(+(\)\(24\!\cdots\!58\)\( + \)\(76\!\cdots\!68\)\( \beta_{1} + \)\(11\!\cdots\!26\)\( \beta_{2} + \)\(77\!\cdots\!42\)\( \beta_{3} + \)\(23\!\cdots\!48\)\( \beta_{4} + \)\(25\!\cdots\!00\)\( \beta_{5} + 87621594645561984000 \beta_{6}) q^{20}\) \(+(-\)\(12\!\cdots\!48\)\( - \)\(12\!\cdots\!52\)\( \beta_{1} - \)\(27\!\cdots\!32\)\( \beta_{2} - \)\(84\!\cdots\!68\)\( \beta_{3} + \)\(11\!\cdots\!20\)\( \beta_{4} + \)\(19\!\cdots\!00\)\( \beta_{5} - \)\(94\!\cdots\!00\)\( \beta_{6}) q^{21}\) \(+(-\)\(81\!\cdots\!43\)\( + \)\(59\!\cdots\!74\)\( \beta_{1} - \)\(21\!\cdots\!82\)\( \beta_{2} - \)\(46\!\cdots\!18\)\( \beta_{3} - \)\(16\!\cdots\!32\)\( \beta_{4} - \)\(43\!\cdots\!53\)\( \beta_{5} + \)\(90\!\cdots\!92\)\( \beta_{6}) q^{22}\) \(+(-\)\(16\!\cdots\!03\)\( + \)\(95\!\cdots\!33\)\( \beta_{1} + \)\(26\!\cdots\!36\)\( \beta_{2} + \)\(16\!\cdots\!65\)\( \beta_{3} + \)\(13\!\cdots\!35\)\( \beta_{4} + \)\(43\!\cdots\!15\)\( \beta_{5} - \)\(75\!\cdots\!85\)\( \beta_{6}) q^{23}\) \(+(\)\(14\!\cdots\!76\)\( + \)\(61\!\cdots\!52\)\( \beta_{1} + \)\(10\!\cdots\!92\)\( \beta_{2} + \)\(16\!\cdots\!76\)\( \beta_{3} - \)\(22\!\cdots\!60\)\( \beta_{4} - \)\(27\!\cdots\!44\)\( \beta_{5} + \)\(55\!\cdots\!00\)\( \beta_{6}) q^{24}\) \(+(-\)\(30\!\cdots\!25\)\( + \)\(18\!\cdots\!00\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} - \)\(95\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!00\)\( \beta_{4} + \)\(99\!\cdots\!00\)\( \beta_{5} - \)\(36\!\cdots\!00\)\( \beta_{6}) q^{25}\) \(+(\)\(36\!\cdots\!10\)\( + \)\(39\!\cdots\!34\)\( \beta_{1} - \)\(11\!\cdots\!96\)\( \beta_{2} - \)\(37\!\cdots\!80\)\( \beta_{3} + \)\(31\!\cdots\!20\)\( \beta_{4} + \)\(15\!\cdots\!48\)\( \beta_{5} + \)\(20\!\cdots\!00\)\( \beta_{6}) q^{26}\) \(+(\)\(16\!\cdots\!54\)\( + \)\(15\!\cdots\!16\)\( \beta_{1} - \)\(23\!\cdots\!06\)\( \beta_{2} + \)\(32\!\cdots\!38\)\( \beta_{3} - \)\(61\!\cdots\!38\)\( \beta_{4} - \)\(31\!\cdots\!42\)\( \beta_{5} - \)\(10\!\cdots\!62\)\( \beta_{6}) q^{27}\) \(+(\)\(60\!\cdots\!56\)\( + \)\(11\!\cdots\!88\)\( \beta_{1} + \)\(57\!\cdots\!72\)\( \beta_{2} + \)\(84\!\cdots\!32\)\( \beta_{3} - \)\(59\!\cdots\!32\)\( \beta_{4} + \)\(23\!\cdots\!12\)\( \beta_{5} + \)\(46\!\cdots\!32\)\( \beta_{6}) q^{28}\) \(+(-\)\(20\!\cdots\!33\)\( + \)\(57\!\cdots\!32\)\( \beta_{1} + \)\(18\!\cdots\!12\)\( \beta_{2} - \)\(12\!\cdots\!01\)\( \beta_{3} + \)\(48\!\cdots\!55\)\( \beta_{4} - \)\(85\!\cdots\!48\)\( \beta_{5} - \)\(16\!\cdots\!00\)\( \beta_{6}) q^{29}\) \(+(\)\(97\!\cdots\!98\)\( + \)\(43\!\cdots\!08\)\( \beta_{1} + \)\(10\!\cdots\!56\)\( \beta_{2} + \)\(17\!\cdots\!52\)\( \beta_{3} - \)\(11\!\cdots\!12\)\( \beta_{4} + \)\(30\!\cdots\!50\)\( \beta_{5} + \)\(50\!\cdots\!00\)\( \beta_{6}) q^{30}\) \(+(\)\(97\!\cdots\!56\)\( - \)\(26\!\cdots\!24\)\( \beta_{1} - \)\(40\!\cdots\!24\)\( \beta_{2} + \)\(19\!\cdots\!36\)\( \beta_{3} - \)\(43\!\cdots\!00\)\( \beta_{4} + \)\(18\!\cdots\!64\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6}) q^{31}\) \(+(-\)\(26\!\cdots\!32\)\( - \)\(32\!\cdots\!44\)\( \beta_{1} - \)\(38\!\cdots\!64\)\( \beta_{2} - \)\(74\!\cdots\!96\)\( \beta_{3} + \)\(38\!\cdots\!96\)\( \beta_{4} - \)\(13\!\cdots\!76\)\( \beta_{5} + \)\(85\!\cdots\!64\)\( \beta_{6}) q^{32}\) \(+(-\)\(12\!\cdots\!26\)\( - \)\(83\!\cdots\!20\)\( \beta_{1} + \)\(10\!\cdots\!12\)\( \beta_{2} - \)\(92\!\cdots\!74\)\( \beta_{3} - \)\(83\!\cdots\!26\)\( \beta_{4} + \)\(55\!\cdots\!76\)\( \beta_{5} + \)\(39\!\cdots\!36\)\( \beta_{6}) q^{33}\) \(+(\)\(44\!\cdots\!70\)\( - \)\(15\!\cdots\!98\)\( \beta_{1} + \)\(29\!\cdots\!32\)\( \beta_{2} + \)\(69\!\cdots\!36\)\( \beta_{3} - \)\(23\!\cdots\!20\)\( \beta_{4} - \)\(14\!\cdots\!24\)\( \beta_{5} - \)\(17\!\cdots\!00\)\( \beta_{6}) q^{34}\) \(+(\)\(47\!\cdots\!68\)\( + \)\(72\!\cdots\!28\)\( \beta_{1} + \)\(32\!\cdots\!96\)\( \beta_{2} + \)\(16\!\cdots\!32\)\( \beta_{3} + \)\(17\!\cdots\!08\)\( \beta_{4} + \)\(22\!\cdots\!00\)\( \beta_{5} + \)\(18\!\cdots\!00\)\( \beta_{6}) q^{35}\) \(+(-\)\(23\!\cdots\!43\)\( + \)\(45\!\cdots\!72\)\( \beta_{1} - \)\(58\!\cdots\!33\)\( \beta_{2} - \)\(12\!\cdots\!79\)\( \beta_{3} - \)\(24\!\cdots\!80\)\( \beta_{4} - \)\(32\!\cdots\!64\)\( \beta_{5} + \)\(67\!\cdots\!00\)\( \beta_{6}) q^{36}\) \(+(-\)\(77\!\cdots\!97\)\( + \)\(14\!\cdots\!84\)\( \beta_{1} - \)\(60\!\cdots\!64\)\( \beta_{2} - \)\(33\!\cdots\!49\)\( \beta_{3} - \)\(11\!\cdots\!01\)\( \beta_{4} + \)\(29\!\cdots\!16\)\( \beta_{5} + \)\(29\!\cdots\!76\)\( \beta_{6}) q^{37}\) \(+(\)\(64\!\cdots\!47\)\( + \)\(10\!\cdots\!26\)\( \beta_{1} + \)\(21\!\cdots\!22\)\( \beta_{2} + \)\(95\!\cdots\!46\)\( \beta_{3} + \)\(40\!\cdots\!04\)\( \beta_{4} - \)\(21\!\cdots\!19\)\( \beta_{5} - \)\(36\!\cdots\!84\)\( \beta_{6}) q^{38}\) \(+(-\)\(75\!\cdots\!87\)\( - \)\(75\!\cdots\!75\)\( \beta_{1} + \)\(25\!\cdots\!60\)\( \beta_{2} + \)\(14\!\cdots\!41\)\( \beta_{3} + \)\(10\!\cdots\!35\)\( \beta_{4} + \)\(85\!\cdots\!87\)\( \beta_{5} + \)\(25\!\cdots\!75\)\( \beta_{6}) q^{39}\) \(+(-\)\(51\!\cdots\!80\)\( - \)\(56\!\cdots\!80\)\( \beta_{1} - \)\(39\!\cdots\!60\)\( \beta_{2} - \)\(13\!\cdots\!20\)\( \beta_{3} - \)\(99\!\cdots\!80\)\( \beta_{4} - \)\(15\!\cdots\!00\)\( \beta_{5} - \)\(90\!\cdots\!00\)\( \beta_{6}) q^{40}\) \(+(-\)\(93\!\cdots\!34\)\( - \)\(93\!\cdots\!84\)\( \beta_{1} - \)\(18\!\cdots\!04\)\( \beta_{2} - \)\(58\!\cdots\!84\)\( \beta_{3} + \)\(28\!\cdots\!80\)\( \beta_{4} - \)\(22\!\cdots\!96\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6}) q^{41}\) \(+(\)\(17\!\cdots\!92\)\( + \)\(22\!\cdots\!40\)\( \beta_{1} + \)\(21\!\cdots\!48\)\( \beta_{2} + \)\(16\!\cdots\!16\)\( \beta_{3} - \)\(37\!\cdots\!16\)\( \beta_{4} + \)\(25\!\cdots\!96\)\( \beta_{5} + \)\(50\!\cdots\!56\)\( \beta_{6}) q^{42}\) \(+(\)\(45\!\cdots\!24\)\( + \)\(35\!\cdots\!75\)\( \beta_{1} + \)\(14\!\cdots\!13\)\( \beta_{2} - \)\(19\!\cdots\!40\)\( \beta_{3} + \)\(80\!\cdots\!40\)\( \beta_{4} - \)\(79\!\cdots\!60\)\( \beta_{5} - \)\(40\!\cdots\!60\)\( \beta_{6}) q^{43}\) \(+(-\)\(22\!\cdots\!60\)\( + \)\(35\!\cdots\!92\)\( \beta_{1} + \)\(41\!\cdots\!92\)\( \beta_{2} - \)\(64\!\cdots\!64\)\( \beta_{3} - \)\(58\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!56\)\( \beta_{5} + \)\(12\!\cdots\!00\)\( \beta_{6}) q^{44}\) \(+(-\)\(68\!\cdots\!73\)\( + \)\(26\!\cdots\!92\)\( \beta_{1} - \)\(44\!\cdots\!56\)\( \beta_{2} + \)\(59\!\cdots\!23\)\( \beta_{3} + \)\(22\!\cdots\!87\)\( \beta_{4} + \)\(13\!\cdots\!00\)\( \beta_{5} - \)\(15\!\cdots\!00\)\( \beta_{6}) q^{45}\) \(+(-\)\(80\!\cdots\!54\)\( - \)\(97\!\cdots\!24\)\( \beta_{1} - \)\(35\!\cdots\!44\)\( \beta_{2} + \)\(41\!\cdots\!36\)\( \beta_{3} - \)\(34\!\cdots\!20\)\( \beta_{4} - \)\(81\!\cdots\!86\)\( \beta_{5} - \)\(55\!\cdots\!00\)\( \beta_{6}) q^{46}\) \(+(-\)\(84\!\cdots\!54\)\( - \)\(31\!\cdots\!42\)\( \beta_{1} + \)\(26\!\cdots\!44\)\( \beta_{2} + \)\(53\!\cdots\!30\)\( \beta_{3} - \)\(28\!\cdots\!30\)\( \beta_{4} + \)\(11\!\cdots\!70\)\( \beta_{5} + \)\(36\!\cdots\!70\)\( \beta_{6}) q^{47}\) \(+(-\)\(54\!\cdots\!24\)\( - \)\(20\!\cdots\!24\)\( \beta_{1} + \)\(97\!\cdots\!60\)\( \beta_{2} - \)\(69\!\cdots\!84\)\( \beta_{3} + \)\(15\!\cdots\!84\)\( \beta_{4} + \)\(33\!\cdots\!96\)\( \beta_{5} - \)\(92\!\cdots\!44\)\( \beta_{6}) q^{48}\) \(+(-\)\(44\!\cdots\!47\)\( + \)\(48\!\cdots\!24\)\( \beta_{1} + \)\(33\!\cdots\!44\)\( \beta_{2} + \)\(96\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!80\)\( \beta_{4} + \)\(10\!\cdots\!96\)\( \beta_{5} + \)\(54\!\cdots\!00\)\( \beta_{6}) q^{49}\) \(+(-\)\(15\!\cdots\!75\)\( + \)\(92\!\cdots\!25\)\( \beta_{1} - \)\(59\!\cdots\!00\)\( \beta_{2} + \)\(42\!\cdots\!00\)\( \beta_{3} - \)\(26\!\cdots\!00\)\( \beta_{4} - \)\(27\!\cdots\!00\)\( \beta_{5} + \)\(50\!\cdots\!00\)\( \beta_{6}) q^{50}\) \(+(-\)\(63\!\cdots\!62\)\( + \)\(17\!\cdots\!40\)\( \beta_{1} - \)\(24\!\cdots\!30\)\( \beta_{2} + \)\(56\!\cdots\!78\)\( \beta_{3} - \)\(26\!\cdots\!70\)\( \beta_{4} + \)\(80\!\cdots\!26\)\( \beta_{5} - \)\(21\!\cdots\!50\)\( \beta_{6}) q^{51}\) \(+(-\)\(27\!\cdots\!42\)\( + \)\(22\!\cdots\!40\)\( \beta_{1} + \)\(21\!\cdots\!46\)\( \beta_{2} - \)\(32\!\cdots\!70\)\( \beta_{3} + \)\(12\!\cdots\!20\)\( \beta_{4} + \)\(53\!\cdots\!20\)\( \beta_{5} + \)\(35\!\cdots\!20\)\( \beta_{6}) q^{52}\) \(+(-\)\(25\!\cdots\!53\)\( - \)\(37\!\cdots\!60\)\( \beta_{1} + \)\(89\!\cdots\!56\)\( \beta_{2} + \)\(37\!\cdots\!83\)\( \beta_{3} - \)\(15\!\cdots\!33\)\( \beta_{4} - \)\(65\!\cdots\!72\)\( \beta_{5} + \)\(23\!\cdots\!08\)\( \beta_{6}) q^{53}\) \(+(-\)\(15\!\cdots\!54\)\( - \)\(35\!\cdots\!60\)\( \beta_{1} + \)\(24\!\cdots\!40\)\( \beta_{2} + \)\(70\!\cdots\!48\)\( \beta_{3} - \)\(19\!\cdots\!00\)\( \beta_{4} + \)\(18\!\cdots\!66\)\( \beta_{5} - \)\(28\!\cdots\!00\)\( \beta_{6}) q^{54}\) \(+(-\)\(17\!\cdots\!57\)\( - \)\(45\!\cdots\!97\)\( \beta_{1} - \)\(19\!\cdots\!04\)\( \beta_{2} + \)\(21\!\cdots\!07\)\( \beta_{3} + \)\(50\!\cdots\!33\)\( \beta_{4} - \)\(11\!\cdots\!75\)\( \beta_{5} + \)\(71\!\cdots\!25\)\( \beta_{6}) q^{55}\) \(+(-\)\(83\!\cdots\!52\)\( - \)\(72\!\cdots\!12\)\( \beta_{1} - \)\(26\!\cdots\!12\)\( \beta_{2} - \)\(93\!\cdots\!64\)\( \beta_{3} + \)\(68\!\cdots\!00\)\( \beta_{4} - \)\(51\!\cdots\!92\)\( \beta_{5} - \)\(45\!\cdots\!00\)\( \beta_{6}) q^{56}\) \(+(-\)\(94\!\cdots\!06\)\( + \)\(14\!\cdots\!04\)\( \beta_{1} - \)\(55\!\cdots\!36\)\( \beta_{2} + \)\(73\!\cdots\!90\)\( \beta_{3} - \)\(60\!\cdots\!90\)\( \beta_{4} + \)\(13\!\cdots\!60\)\( \beta_{5} - \)\(25\!\cdots\!40\)\( \beta_{6}) q^{57}\) \(+(-\)\(44\!\cdots\!14\)\( + \)\(10\!\cdots\!30\)\( \beta_{1} + \)\(94\!\cdots\!96\)\( \beta_{2} + \)\(98\!\cdots\!16\)\( \beta_{3} - \)\(12\!\cdots\!16\)\( \beta_{4} - \)\(78\!\cdots\!64\)\( \beta_{5} + \)\(90\!\cdots\!96\)\( \beta_{6}) q^{58}\) \(+(-\)\(12\!\cdots\!80\)\( - \)\(40\!\cdots\!97\)\( \beta_{1} + \)\(87\!\cdots\!73\)\( \beta_{2} + \)\(18\!\cdots\!52\)\( \beta_{3} + \)\(32\!\cdots\!20\)\( \beta_{4} - \)\(92\!\cdots\!00\)\( \beta_{5} - \)\(11\!\cdots\!00\)\( \beta_{6}) q^{59}\) \(+(-\)\(40\!\cdots\!12\)\( - \)\(16\!\cdots\!52\)\( \beta_{1} - \)\(26\!\cdots\!64\)\( \beta_{2} - \)\(83\!\cdots\!88\)\( \beta_{3} - \)\(48\!\cdots\!72\)\( \beta_{4} - \)\(37\!\cdots\!00\)\( \beta_{5} - \)\(12\!\cdots\!00\)\( \beta_{6}) q^{60}\) \(+(\)\(12\!\cdots\!87\)\( - \)\(46\!\cdots\!00\)\( \beta_{1} + \)\(54\!\cdots\!60\)\( \beta_{2} - \)\(80\!\cdots\!45\)\( \beta_{3} - \)\(88\!\cdots\!65\)\( \beta_{4} + \)\(16\!\cdots\!60\)\( \beta_{5} + \)\(78\!\cdots\!00\)\( \beta_{6}) q^{61}\) \(+(\)\(24\!\cdots\!28\)\( - \)\(13\!\cdots\!04\)\( \beta_{1} - \)\(15\!\cdots\!96\)\( \beta_{2} + \)\(38\!\cdots\!56\)\( \beta_{3} + \)\(50\!\cdots\!44\)\( \beta_{4} + \)\(11\!\cdots\!36\)\( \beta_{5} - \)\(13\!\cdots\!04\)\( \beta_{6}) q^{62}\) \(+(\)\(85\!\cdots\!07\)\( - \)\(12\!\cdots\!13\)\( \beta_{1} + \)\(13\!\cdots\!16\)\( \beta_{2} - \)\(24\!\cdots\!09\)\( \beta_{3} + \)\(15\!\cdots\!09\)\( \beta_{4} - \)\(15\!\cdots\!19\)\( \beta_{5} + \)\(12\!\cdots\!41\)\( \beta_{6}) q^{63}\) \(+(\)\(20\!\cdots\!24\)\( + \)\(57\!\cdots\!76\)\( \beta_{1} + \)\(80\!\cdots\!16\)\( \beta_{2} + \)\(12\!\cdots\!20\)\( \beta_{3} + \)\(40\!\cdots\!40\)\( \beta_{4} + \)\(27\!\cdots\!52\)\( \beta_{5} + \)\(45\!\cdots\!00\)\( \beta_{6}) q^{64}\) \(+(\)\(18\!\cdots\!64\)\( + \)\(22\!\cdots\!44\)\( \beta_{1} + \)\(28\!\cdots\!08\)\( \beta_{2} - \)\(50\!\cdots\!64\)\( \beta_{3} - \)\(22\!\cdots\!16\)\( \beta_{4} + \)\(11\!\cdots\!00\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6}) q^{65}\) \(+(\)\(82\!\cdots\!32\)\( + \)\(17\!\cdots\!00\)\( \beta_{1} - \)\(40\!\cdots\!60\)\( \beta_{2} + \)\(31\!\cdots\!04\)\( \beta_{3} + \)\(16\!\cdots\!40\)\( \beta_{4} - \)\(11\!\cdots\!72\)\( \beta_{5} + \)\(77\!\cdots\!00\)\( \beta_{6}) q^{66}\) \(+(\)\(83\!\cdots\!06\)\( + \)\(74\!\cdots\!19\)\( \beta_{1} + \)\(18\!\cdots\!99\)\( \beta_{2} + \)\(53\!\cdots\!94\)\( \beta_{3} + \)\(42\!\cdots\!06\)\( \beta_{4} + \)\(12\!\cdots\!34\)\( \beta_{5} + \)\(13\!\cdots\!74\)\( \beta_{6}) q^{67}\) \(+(\)\(48\!\cdots\!70\)\( - \)\(62\!\cdots\!08\)\( \beta_{1} - \)\(67\!\cdots\!54\)\( \beta_{2} + \)\(13\!\cdots\!02\)\( \beta_{3} - \)\(47\!\cdots\!52\)\( \beta_{4} + \)\(10\!\cdots\!32\)\( \beta_{5} - \)\(49\!\cdots\!48\)\( \beta_{6}) q^{68}\) \(+(-\)\(84\!\cdots\!80\)\( - \)\(12\!\cdots\!28\)\( \beta_{1} + \)\(32\!\cdots\!72\)\( \beta_{2} - \)\(29\!\cdots\!80\)\( \beta_{3} - \)\(46\!\cdots\!00\)\( \beta_{4} - \)\(16\!\cdots\!96\)\( \beta_{5} + \)\(73\!\cdots\!00\)\( \beta_{6}) q^{69}\) \(+(-\)\(69\!\cdots\!44\)\( - \)\(16\!\cdots\!24\)\( \beta_{1} - \)\(24\!\cdots\!68\)\( \beta_{2} - \)\(85\!\cdots\!56\)\( \beta_{3} - \)\(23\!\cdots\!64\)\( \beta_{4} - \)\(58\!\cdots\!00\)\( \beta_{5} - \)\(38\!\cdots\!00\)\( \beta_{6}) q^{70}\) \(+(-\)\(78\!\cdots\!33\)\( + \)\(11\!\cdots\!75\)\( \beta_{1} + \)\(47\!\cdots\!80\)\( \beta_{2} + \)\(16\!\cdots\!15\)\( \beta_{3} + \)\(10\!\cdots\!05\)\( \beta_{4} + \)\(93\!\cdots\!05\)\( \beta_{5} - \)\(11\!\cdots\!75\)\( \beta_{6}) q^{71}\) \(+(-\)\(27\!\cdots\!36\)\( + \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(18\!\cdots\!96\)\( \beta_{2} + \)\(70\!\cdots\!32\)\( \beta_{3} - \)\(53\!\cdots\!32\)\( \beta_{4} + \)\(20\!\cdots\!92\)\( \beta_{5} + \)\(41\!\cdots\!12\)\( \beta_{6}) q^{72}\) \(+(-\)\(28\!\cdots\!84\)\( + \)\(93\!\cdots\!36\)\( \beta_{1} + \)\(62\!\cdots\!80\)\( \beta_{2} + \)\(27\!\cdots\!66\)\( \beta_{3} - \)\(19\!\cdots\!66\)\( \beta_{4} - \)\(65\!\cdots\!24\)\( \beta_{5} - \)\(62\!\cdots\!64\)\( \beta_{6}) q^{73}\) \(+(-\)\(13\!\cdots\!42\)\( + \)\(34\!\cdots\!06\)\( \beta_{1} - \)\(17\!\cdots\!04\)\( \beta_{2} - \)\(14\!\cdots\!84\)\( \beta_{3} + \)\(13\!\cdots\!40\)\( \beta_{4} + \)\(54\!\cdots\!84\)\( \beta_{5} + \)\(90\!\cdots\!00\)\( \beta_{6}) q^{74}\) \(+(-\)\(64\!\cdots\!00\)\( - \)\(31\!\cdots\!25\)\( \beta_{1} + \)\(13\!\cdots\!25\)\( \beta_{2} + \)\(28\!\cdots\!00\)\( \beta_{3} + \)\(12\!\cdots\!00\)\( \beta_{4} - \)\(53\!\cdots\!00\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{75}\) \(+(-\)\(89\!\cdots\!24\)\( - \)\(86\!\cdots\!84\)\( \beta_{1} - \)\(71\!\cdots\!84\)\( \beta_{2} + \)\(17\!\cdots\!72\)\( \beta_{3} + \)\(14\!\cdots\!00\)\( \beta_{4} + \)\(16\!\cdots\!96\)\( \beta_{5} - \)\(25\!\cdots\!00\)\( \beta_{6}) q^{76}\) \(+(\)\(19\!\cdots\!72\)\( - \)\(81\!\cdots\!88\)\( \beta_{1} + \)\(64\!\cdots\!48\)\( \beta_{2} + \)\(42\!\cdots\!24\)\( \beta_{3} - \)\(38\!\cdots\!24\)\( \beta_{4} + \)\(40\!\cdots\!44\)\( \beta_{5} + \)\(66\!\cdots\!84\)\( \beta_{6}) q^{77}\) \(+(\)\(73\!\cdots\!26\)\( - \)\(53\!\cdots\!84\)\( \beta_{1} - \)\(83\!\cdots\!36\)\( \beta_{2} - \)\(40\!\cdots\!08\)\( \beta_{3} + \)\(29\!\cdots\!08\)\( \beta_{4} - \)\(29\!\cdots\!38\)\( \beta_{5} + \)\(66\!\cdots\!32\)\( \beta_{6}) q^{78}\) \(+(\)\(38\!\cdots\!66\)\( + \)\(33\!\cdots\!98\)\( \beta_{1} + \)\(54\!\cdots\!48\)\( \beta_{2} - \)\(22\!\cdots\!70\)\( \beta_{3} + \)\(76\!\cdots\!50\)\( \beta_{4} + \)\(45\!\cdots\!86\)\( \beta_{5} + \)\(59\!\cdots\!50\)\( \beta_{6}) q^{79}\) \(+(\)\(39\!\cdots\!44\)\( + \)\(10\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!68\)\( \beta_{2} + \)\(30\!\cdots\!56\)\( \beta_{3} - \)\(48\!\cdots\!36\)\( \beta_{4} + \)\(42\!\cdots\!00\)\( \beta_{5} - \)\(55\!\cdots\!00\)\( \beta_{6}) q^{80}\) \(+(\)\(69\!\cdots\!51\)\( - \)\(73\!\cdots\!36\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} - \)\(30\!\cdots\!94\)\( \beta_{3} + \)\(83\!\cdots\!30\)\( \beta_{4} - \)\(22\!\cdots\!40\)\( \beta_{5} - \)\(46\!\cdots\!00\)\( \beta_{6}) q^{81}\) \(+(\)\(90\!\cdots\!58\)\( + \)\(15\!\cdots\!46\)\( \beta_{1} + \)\(44\!\cdots\!64\)\( \beta_{2} + \)\(14\!\cdots\!56\)\( \beta_{3} - \)\(50\!\cdots\!56\)\( \beta_{4} + \)\(20\!\cdots\!96\)\( \beta_{5} + \)\(11\!\cdots\!56\)\( \beta_{6}) q^{82}\) \(+(-\)\(50\!\cdots\!52\)\( - \)\(27\!\cdots\!31\)\( \beta_{1} - \)\(61\!\cdots\!53\)\( \beta_{2} - \)\(98\!\cdots\!00\)\( \beta_{3} + \)\(40\!\cdots\!00\)\( \beta_{4} + \)\(24\!\cdots\!00\)\( \beta_{5} + \)\(17\!\cdots\!00\)\( \beta_{6}) q^{83}\) \(+(-\)\(13\!\cdots\!64\)\( - \)\(12\!\cdots\!08\)\( \beta_{1} - \)\(92\!\cdots\!68\)\( \beta_{2} - \)\(37\!\cdots\!36\)\( \beta_{3} + \)\(92\!\cdots\!40\)\( \beta_{4} - \)\(57\!\cdots\!48\)\( \beta_{5} - \)\(41\!\cdots\!00\)\( \beta_{6}) q^{84}\) \(+(-\)\(20\!\cdots\!82\)\( - \)\(36\!\cdots\!72\)\( \beta_{1} - \)\(10\!\cdots\!04\)\( \beta_{2} + \)\(23\!\cdots\!82\)\( \beta_{3} - \)\(32\!\cdots\!42\)\( \beta_{4} + \)\(95\!\cdots\!00\)\( \beta_{5} + \)\(37\!\cdots\!00\)\( \beta_{6}) q^{85}\) \(+(-\)\(35\!\cdots\!51\)\( + \)\(84\!\cdots\!34\)\( \beta_{1} + \)\(57\!\cdots\!34\)\( \beta_{2} + \)\(29\!\cdots\!02\)\( \beta_{3} - \)\(21\!\cdots\!00\)\( \beta_{4} + \)\(21\!\cdots\!47\)\( \beta_{5} + \)\(86\!\cdots\!00\)\( \beta_{6}) q^{86}\) \(+(-\)\(59\!\cdots\!35\)\( + \)\(30\!\cdots\!29\)\( \beta_{1} + \)\(22\!\cdots\!80\)\( \beta_{2} + \)\(18\!\cdots\!61\)\( \beta_{3} - \)\(60\!\cdots\!61\)\( \beta_{4} + \)\(81\!\cdots\!11\)\( \beta_{5} - \)\(14\!\cdots\!29\)\( \beta_{6}) q^{87}\) \(+(-\)\(26\!\cdots\!92\)\( + \)\(25\!\cdots\!52\)\( \beta_{1} + \)\(27\!\cdots\!28\)\( \beta_{2} - \)\(34\!\cdots\!84\)\( \beta_{3} + \)\(23\!\cdots\!84\)\( \beta_{4} - \)\(71\!\cdots\!04\)\( \beta_{5} - \)\(19\!\cdots\!44\)\( \beta_{6}) q^{88}\) \(+(-\)\(23\!\cdots\!24\)\( + \)\(69\!\cdots\!96\)\( \beta_{1} - \)\(18\!\cdots\!24\)\( \beta_{2} - \)\(10\!\cdots\!98\)\( \beta_{3} - \)\(89\!\cdots\!70\)\( \beta_{4} - \)\(25\!\cdots\!84\)\( \beta_{5} + \)\(55\!\cdots\!00\)\( \beta_{6}) q^{89}\) \(+(-\)\(21\!\cdots\!66\)\( + \)\(33\!\cdots\!14\)\( \beta_{1} - \)\(31\!\cdots\!52\)\( \beta_{2} + \)\(94\!\cdots\!16\)\( \beta_{3} - \)\(40\!\cdots\!96\)\( \beta_{4} + \)\(26\!\cdots\!00\)\( \beta_{5} + \)\(36\!\cdots\!00\)\( \beta_{6}) q^{90}\) \(+(-\)\(43\!\cdots\!40\)\( - \)\(35\!\cdots\!92\)\( \beta_{1} - \)\(11\!\cdots\!92\)\( \beta_{2} + \)\(55\!\cdots\!16\)\( \beta_{3} + \)\(53\!\cdots\!00\)\( \beta_{4} + \)\(19\!\cdots\!08\)\( \beta_{5} - \)\(19\!\cdots\!00\)\( \beta_{6}) q^{91}\) \(+(\)\(10\!\cdots\!64\)\( - \)\(22\!\cdots\!20\)\( \beta_{1} + \)\(51\!\cdots\!12\)\( \beta_{2} + \)\(15\!\cdots\!56\)\( \beta_{3} - \)\(51\!\cdots\!56\)\( \beta_{4} + \)\(58\!\cdots\!96\)\( \beta_{5} + \)\(23\!\cdots\!56\)\( \beta_{6}) q^{92}\) \(+(\)\(12\!\cdots\!72\)\( - \)\(49\!\cdots\!64\)\( \beta_{1} + \)\(10\!\cdots\!40\)\( \beta_{2} - \)\(26\!\cdots\!92\)\( \beta_{3} + \)\(72\!\cdots\!92\)\( \beta_{4} - \)\(15\!\cdots\!12\)\( \beta_{5} + \)\(58\!\cdots\!68\)\( \beta_{6}) q^{93}\) \(+(\)\(32\!\cdots\!20\)\( + \)\(43\!\cdots\!80\)\( \beta_{1} - \)\(11\!\cdots\!20\)\( \beta_{2} - \)\(13\!\cdots\!32\)\( \beta_{3} + \)\(14\!\cdots\!00\)\( \beta_{4} - \)\(25\!\cdots\!64\)\( \beta_{5} - \)\(27\!\cdots\!00\)\( \beta_{6}) q^{94}\) \(+(\)\(32\!\cdots\!65\)\( + \)\(66\!\cdots\!65\)\( \beta_{1} - \)\(50\!\cdots\!20\)\( \beta_{2} - \)\(47\!\cdots\!15\)\( \beta_{3} - \)\(43\!\cdots\!85\)\( \beta_{4} + \)\(50\!\cdots\!75\)\( \beta_{5} - \)\(15\!\cdots\!25\)\( \beta_{6}) q^{95}\) \(+(\)\(12\!\cdots\!00\)\( + \)\(61\!\cdots\!16\)\( \beta_{1} - \)\(37\!\cdots\!24\)\( \beta_{2} + \)\(16\!\cdots\!68\)\( \beta_{3} - \)\(36\!\cdots\!40\)\( \beta_{4} + \)\(11\!\cdots\!68\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6}) q^{96}\) \(+(\)\(10\!\cdots\!48\)\( + \)\(43\!\cdots\!12\)\( \beta_{1} + \)\(39\!\cdots\!96\)\( \beta_{2} + \)\(15\!\cdots\!58\)\( \beta_{3} + \)\(15\!\cdots\!42\)\( \beta_{4} - \)\(19\!\cdots\!72\)\( \beta_{5} + \)\(38\!\cdots\!08\)\( \beta_{6}) q^{97}\) \(+(-\)\(25\!\cdots\!05\)\( - \)\(25\!\cdots\!05\)\( \beta_{1} - \)\(86\!\cdots\!24\)\( \beta_{2} - \)\(18\!\cdots\!76\)\( \beta_{3} + \)\(44\!\cdots\!76\)\( \beta_{4} - \)\(42\!\cdots\!16\)\( \beta_{5} - \)\(63\!\cdots\!76\)\( \beta_{6}) q^{98}\) \(+(-\)\(28\!\cdots\!56\)\( - \)\(16\!\cdots\!57\)\( \beta_{1} + \)\(30\!\cdots\!93\)\( \beta_{2} + \)\(81\!\cdots\!36\)\( \beta_{3} - \)\(43\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!68\)\( \beta_{5} - \)\(67\!\cdots\!00\)\( \beta_{6}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut -\mathstrut 31407330351408q^{2} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!64\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!04\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!76\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!92\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!71\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut -\mathstrut 31407330351408q^{2} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!64\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!04\)\(q^{4} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!76\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!92\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!20\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!71\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!56\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!28\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!34\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!32\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!32\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!42\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!04\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!80\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!36\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!36\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!04\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!40\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!75\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!24\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!84\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!30\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!04\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!48\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!88\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!28\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!88\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!58\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!48\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!66\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!04\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!56\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!32\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!50\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!76\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!08\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!04\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!01\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!56\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!68\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!14\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!20\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!40\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!60\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!94\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!24\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!96\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!44\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!08\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!92\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!84\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!48\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!76\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!60\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!54\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!52\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!40\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!64\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!92\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!20\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!47\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!04\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!24\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!92\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!76\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!60\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!40\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!90\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!36\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!92\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!92\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!08\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!44\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!42\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!56\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!68\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(1400531600527934473811256\) \(x^{5}\mathstrut +\mathstrut \) \(92429106535860966322690362643440028\) \(x^{4}\mathstrut +\mathstrut \) \(486502004825754823566786579226467181483733375376\) \(x^{3}\mathstrut -\mathstrut \) \(41390338158988484679355574715314473323669246141474080139600\) \(x^{2}\mathstrut -\mathstrut \) \(47785461930919140795588898989186212855196409324706742802409577734342400\) \(x\mathstrut +\mathstrut \) \(5612439960923763868733925256800794059272997589318959539312206365735127554315560000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 21 \)
\(\beta_{2}\)\(=\)\((\)\(13\!\cdots\!73\) \(\nu^{6}\mathstrut -\mathstrut \) \(43\!\cdots\!29\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!94\) \(\nu^{4}\mathstrut +\mathstrut \) \(65\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(24\!\cdots\!28\) \(\nu^{2}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu\mathstrut +\mathstrut \) \(12\!\cdots\!60\)\()/\)\(25\!\cdots\!88\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(15\!\cdots\!61\) \(\nu^{6}\mathstrut +\mathstrut \) \(50\!\cdots\!53\) \(\nu^{5}\mathstrut +\mathstrut \) \(18\!\cdots\!58\) \(\nu^{4}\mathstrut -\mathstrut \) \(75\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(57\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(17\!\cdots\!40\) \(\nu\mathstrut -\mathstrut \) \(10\!\cdots\!64\)\()/\)\(95\!\cdots\!44\)
\(\beta_{4}\)\(=\)\((\)\(18\!\cdots\!99\) \(\nu^{6}\mathstrut +\mathstrut \) \(28\!\cdots\!61\) \(\nu^{5}\mathstrut -\mathstrut \) \(24\!\cdots\!62\) \(\nu^{4}\mathstrut -\mathstrut \) \(21\!\cdots\!12\) \(\nu^{3}\mathstrut +\mathstrut \) \(72\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(53\!\cdots\!80\) \(\nu\mathstrut -\mathstrut \) \(44\!\cdots\!40\)\()/\)\(12\!\cdots\!40\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(33\!\cdots\!43\) \(\nu^{6}\mathstrut +\mathstrut \) \(23\!\cdots\!79\) \(\nu^{5}\mathstrut +\mathstrut \) \(39\!\cdots\!30\) \(\nu^{4}\mathstrut -\mathstrut \) \(30\!\cdots\!64\) \(\nu^{3}\mathstrut -\mathstrut \) \(53\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(55\!\cdots\!68\) \(\nu\mathstrut -\mathstrut \) \(52\!\cdots\!08\)\()/\)\(32\!\cdots\!36\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(37\!\cdots\!79\) \(\nu^{6}\mathstrut -\mathstrut \) \(71\!\cdots\!61\) \(\nu^{5}\mathstrut +\mathstrut \) \(40\!\cdots\!42\) \(\nu^{4}\mathstrut +\mathstrut \) \(81\!\cdots\!72\) \(\nu^{3}\mathstrut -\mathstrut \) \(44\!\cdots\!80\) \(\nu^{2}\mathstrut -\mathstrut \) \(33\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(44\!\cdots\!80\)\()/\)\(32\!\cdots\!60\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(21\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(31239\) \(\beta_{2}\mathstrut -\mathstrut \) \(4751692611940\) \(\beta_{1}\mathstrut +\mathstrut \) \(921949945033243971463488761\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(42\) \(\beta_{6}\mathstrut +\mathstrut \) \(8322\) \(\beta_{5}\mathstrut -\mathstrut \) \(10260742\) \(\beta_{4}\mathstrut -\mathstrut \) \(535021437249\) \(\beta_{3}\mathstrut +\mathstrut \) \(268434092474822827\) \(\beta_{2}\mathstrut +\mathstrut \) \(101634125699569034147392446\) \(\beta_{1}\mathstrut -\mathstrut \) \(273801421862883403640690762869661130673\)\()/6912\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(42626265665746\) \(\beta_{6}\mathstrut +\mathstrut \) \(5454036319681910\) \(\beta_{5}\mathstrut -\mathstrut \) \(67267238634623728354\) \(\beta_{4}\mathstrut +\mathstrut \) \(9562404747689700160077795\) \(\beta_{3}\mathstrut +\mathstrut \) \(498401500817003729460333517819\) \(\beta_{2}\mathstrut -\mathstrut \) \(67905865059166950371798663289432229934\) \(\beta_{1}\mathstrut +\mathstrut \) \(5856348533646391890503895030399830146824500703713379\)\()/20736\)
\(\nu^{5}\)\(=\)\((\)\(143620980826951946467694594\) \(\beta_{6}\mathstrut +\mathstrut \) \(32927949678333827344996394234\) \(\beta_{5}\mathstrut -\mathstrut \) \(30181395137311732663047694536814\) \(\beta_{4}\mathstrut -\mathstrut \) \(2811740194972628139255298324010469411\) \(\beta_{3}\mathstrut +\mathstrut \) \(961431947010529603301362293785252910285109\) \(\beta_{2}\mathstrut +\mathstrut \) \(265028194601956683646060279747275536227588528802782\) \(\beta_{1}\mathstrut -\mathstrut \) \(1304287680774917922401743805934982636589412503275002232600736739\)\()/20736\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(21115822231702659962369104000132091542\) \(\beta_{6}\mathstrut +\mathstrut \) \(1654060343202791154047339733379286645954\) \(\beta_{5}\mathstrut -\mathstrut \) \(24251169340518390874806368246168395048347078\) \(\beta_{4}\mathstrut +\mathstrut \) \(3109350977751397774901165596321092470351213795361\) \(\beta_{3}\mathstrut +\mathstrut \) \(163364834885662550659605149686597478065533847598162073\) \(\beta_{2}\mathstrut -\mathstrut \) \(30219766887836621982296293647198921627296282012133903160308362\) \(\beta_{1}\mathstrut +\mathstrut \) \(1696824508449597394032067197885930670451958422815631610201782091796588158817\)\()/6912\)

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
9.08809e11
5.71170e11
3.74539e11
1.22894e11
−4.68207e11
−4.94369e11
−1.01484e12
−4.81096e13 −1.58991e21 1.69556e27 1.07790e31 7.64900e34 5.70296e37 −5.17945e40 −3.81503e41 −5.18572e44
1.2 −3.19029e13 3.23465e21 3.98828e26 1.00222e30 −1.03195e35 1.60489e37 7.02319e39 7.55364e42 −3.19739e43
1.3 −2.24646e13 −6.44834e20 −1.14310e26 −1.81181e31 1.44860e34 −2.76134e37 1.64729e40 −2.49351e42 4.07017e44
1.4 −1.03857e13 −5.04839e20 −5.11108e26 2.06848e31 5.24310e33 −1.95248e37 1.17366e40 −2.65446e42 −2.14826e44
1.5 1.79872e13 −3.11193e21 −2.95431e26 −4.75717e30 −5.59749e34 3.90338e37 −1.64475e40 6.77478e42 −8.55680e43
1.6 1.92430e13 1.59613e21 −2.48679e26 −3.02831e30 3.07142e34 1.70526e37 −1.66961e40 −3.61697e41 −5.82737e43
1.7 4.42254e13 −3.38901e20 1.33691e27 3.67006e30 −1.49880e34 −4.35255e37 3.17513e40 −2.79447e42 1.62310e44
\(n\): 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 \(S_{90}^{\mathrm{new}}(\Gamma_0(1))\).