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.

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

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

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

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 \(S_{90}^{\mathrm{new}}(\Gamma_0(1))\).