Properties

Label 2.82.a.b
Level 2
Weight 82
Character orbit 2.a
Self dual Yes
Analytic conductor 83.100
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 82 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(83.1002571076\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{15}\cdot 5^{5}\cdot 7^{2}\cdot 11 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +1099511627776 q^{2} +(-1777934344659239676 - \beta_{1}) q^{3} +\)\(12\!\cdots\!76\)\( q^{4} +(\)\(62\!\cdots\!70\)\( + 181291919 \beta_{1} + \beta_{2}) q^{5} +(-\)\(19\!\cdots\!76\)\( - 1099511627776 \beta_{1}) q^{6} +(\)\(45\!\cdots\!48\)\( - 6509551465304 \beta_{1} - 103354 \beta_{2} - \beta_{3}) q^{7} +\)\(13\!\cdots\!76\)\( q^{8} +(\)\(25\!\cdots\!73\)\( + 15058434447394240618 \beta_{1} + 12455607542 \beta_{2} - 16652 \beta_{3}) q^{9} +O(q^{10})\) \( q +1099511627776 q^{2} +(-1777934344659239676 - \beta_{1}) q^{3} +\)\(12\!\cdots\!76\)\( q^{4} +(\)\(62\!\cdots\!70\)\( + 181291919 \beta_{1} + \beta_{2}) q^{5} +(-\)\(19\!\cdots\!76\)\( - 1099511627776 \beta_{1}) q^{6} +(\)\(45\!\cdots\!48\)\( - 6509551465304 \beta_{1} - 103354 \beta_{2} - \beta_{3}) q^{7} +\)\(13\!\cdots\!76\)\( q^{8} +(\)\(25\!\cdots\!73\)\( + 15058434447394240618 \beta_{1} + 12455607542 \beta_{2} - 16652 \beta_{3}) q^{9} +(\)\(68\!\cdots\!20\)\( + \)\(19\!\cdots\!44\)\( \beta_{1} + 1099511627776 \beta_{2}) q^{10} +(\)\(82\!\cdots\!72\)\( + \)\(23\!\cdots\!21\)\( \beta_{1} - 34705923827652 \beta_{2} - 61258538 \beta_{3}) q^{11} +(-\)\(21\!\cdots\!76\)\( - \)\(12\!\cdots\!76\)\( \beta_{1}) q^{12} +(\)\(14\!\cdots\!54\)\( - \)\(71\!\cdots\!05\)\( \beta_{1} - 1277062033641831 \beta_{2} - 56391928264 \beta_{3}) q^{13} +(\)\(50\!\cdots\!48\)\( - \)\(71\!\cdots\!04\)\( \beta_{1} - 113638924777160704 \beta_{2} - 1099511627776 \beta_{3}) q^{14} +(-\)\(13\!\cdots\!20\)\( - \)\(18\!\cdots\!24\)\( \beta_{1} + 8521971385661972154 \beta_{2} + 23670517298625 \beta_{3}) q^{15} +\)\(14\!\cdots\!76\)\( q^{16} +(\)\(42\!\cdots\!78\)\( + \)\(37\!\cdots\!06\)\( \beta_{1} + \)\(30\!\cdots\!18\)\( \beta_{2} - 420103945989308 \beta_{3}) q^{17} +(\)\(27\!\cdots\!48\)\( + \)\(16\!\cdots\!68\)\( \beta_{1} + \)\(13\!\cdots\!92\)\( \beta_{2} - 18309067625725952 \beta_{3}) q^{18} +(\)\(14\!\cdots\!80\)\( + \)\(35\!\cdots\!03\)\( \beta_{1} + \)\(15\!\cdots\!12\)\( \beta_{2} + 162750783707526378 \beta_{3}) q^{19} +(\)\(75\!\cdots\!20\)\( + \)\(21\!\cdots\!44\)\( \beta_{1} + \)\(12\!\cdots\!76\)\( \beta_{2}) q^{20} +(-\)\(36\!\cdots\!48\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(14\!\cdots\!72\)\( \beta_{2} + 590548218301233432 \beta_{3}) q^{21} +(\)\(90\!\cdots\!72\)\( + \)\(25\!\cdots\!96\)\( \beta_{1} - \)\(38\!\cdots\!52\)\( \beta_{2} - 67354474831557951488 \beta_{3}) q^{22} +(\)\(49\!\cdots\!44\)\( + \)\(22\!\cdots\!12\)\( \beta_{1} + \)\(51\!\cdots\!86\)\( \beta_{2} + \)\(42\!\cdots\!09\)\( \beta_{3}) q^{23} +(-\)\(23\!\cdots\!76\)\( - \)\(13\!\cdots\!76\)\( \beta_{1}) q^{24} +(\)\(18\!\cdots\!75\)\( - \)\(35\!\cdots\!80\)\( \beta_{1} + \)\(13\!\cdots\!80\)\( \beta_{2} - \)\(30\!\cdots\!00\)\( \beta_{3}) q^{25} +(\)\(15\!\cdots\!04\)\( - \)\(79\!\cdots\!80\)\( \beta_{1} - \)\(14\!\cdots\!56\)\( \beta_{2} - \)\(62\!\cdots\!64\)\( \beta_{3}) q^{26} +(-\)\(10\!\cdots\!20\)\( - \)\(27\!\cdots\!50\)\( \beta_{1} - \)\(28\!\cdots\!36\)\( \beta_{2} + \)\(55\!\cdots\!66\)\( \beta_{3}) q^{27} +(\)\(55\!\cdots\!48\)\( - \)\(78\!\cdots\!04\)\( \beta_{1} - \)\(12\!\cdots\!04\)\( \beta_{2} - \)\(12\!\cdots\!76\)\( \beta_{3}) q^{28} +(\)\(22\!\cdots\!50\)\( - \)\(90\!\cdots\!45\)\( \beta_{1} + \)\(10\!\cdots\!97\)\( \beta_{2} - \)\(41\!\cdots\!32\)\( \beta_{3}) q^{29} +(-\)\(15\!\cdots\!20\)\( - \)\(20\!\cdots\!24\)\( \beta_{1} + \)\(93\!\cdots\!04\)\( \beta_{2} + \)\(26\!\cdots\!00\)\( \beta_{3}) q^{30} +(\)\(11\!\cdots\!52\)\( - \)\(66\!\cdots\!92\)\( \beta_{1} + \)\(19\!\cdots\!28\)\( \beta_{2} - \)\(35\!\cdots\!68\)\( \beta_{3}) q^{31} +\)\(16\!\cdots\!76\)\( q^{32} +(-\)\(17\!\cdots\!72\)\( - \)\(13\!\cdots\!10\)\( \beta_{1} - \)\(15\!\cdots\!02\)\( \beta_{2} - \)\(16\!\cdots\!88\)\( \beta_{3}) q^{33} +(\)\(46\!\cdots\!28\)\( + \)\(40\!\cdots\!56\)\( \beta_{1} + \)\(33\!\cdots\!68\)\( \beta_{2} - \)\(46\!\cdots\!08\)\( \beta_{3}) q^{34} +(-\)\(27\!\cdots\!40\)\( + \)\(13\!\cdots\!52\)\( \beta_{1} + \)\(28\!\cdots\!08\)\( \beta_{2} + \)\(87\!\cdots\!00\)\( \beta_{3}) q^{35} +(\)\(30\!\cdots\!48\)\( + \)\(18\!\cdots\!68\)\( \beta_{1} + \)\(15\!\cdots\!92\)\( \beta_{2} - \)\(20\!\cdots\!52\)\( \beta_{3}) q^{36} +(\)\(71\!\cdots\!98\)\( + \)\(86\!\cdots\!51\)\( \beta_{1} - \)\(10\!\cdots\!07\)\( \beta_{2} - \)\(38\!\cdots\!08\)\( \beta_{3}) q^{37} +(\)\(15\!\cdots\!80\)\( + \)\(39\!\cdots\!28\)\( \beta_{1} + \)\(16\!\cdots\!12\)\( \beta_{2} + \)\(17\!\cdots\!28\)\( \beta_{3}) q^{38} +(\)\(47\!\cdots\!96\)\( - \)\(51\!\cdots\!12\)\( \beta_{1} - \)\(70\!\cdots\!50\)\( \beta_{2} + \)\(13\!\cdots\!25\)\( \beta_{3}) q^{39} +(\)\(82\!\cdots\!20\)\( + \)\(24\!\cdots\!44\)\( \beta_{1} + \)\(13\!\cdots\!76\)\( \beta_{2}) q^{40} +(\)\(60\!\cdots\!22\)\( - \)\(34\!\cdots\!96\)\( \beta_{1} + \)\(64\!\cdots\!76\)\( \beta_{2} - \)\(34\!\cdots\!56\)\( \beta_{3}) q^{41} +(-\)\(40\!\cdots\!48\)\( - \)\(13\!\cdots\!92\)\( \beta_{1} - \)\(16\!\cdots\!72\)\( \beta_{2} + \)\(64\!\cdots\!32\)\( \beta_{3}) q^{42} +(\)\(97\!\cdots\!04\)\( - \)\(27\!\cdots\!15\)\( \beta_{1} - \)\(64\!\cdots\!64\)\( \beta_{2} + \)\(46\!\cdots\!84\)\( \beta_{3}) q^{43} +(\)\(99\!\cdots\!72\)\( + \)\(27\!\cdots\!96\)\( \beta_{1} - \)\(41\!\cdots\!52\)\( \beta_{2} - \)\(74\!\cdots\!88\)\( \beta_{3}) q^{44} +(\)\(10\!\cdots\!10\)\( + \)\(42\!\cdots\!47\)\( \beta_{1} + \)\(20\!\cdots\!13\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{45} +(\)\(54\!\cdots\!44\)\( + \)\(25\!\cdots\!12\)\( \beta_{1} + \)\(56\!\cdots\!36\)\( \beta_{2} + \)\(46\!\cdots\!84\)\( \beta_{3}) q^{46} +(\)\(17\!\cdots\!88\)\( + \)\(44\!\cdots\!04\)\( \beta_{1} + \)\(92\!\cdots\!32\)\( \beta_{2} + \)\(47\!\cdots\!58\)\( \beta_{3}) q^{47} +(-\)\(25\!\cdots\!76\)\( - \)\(14\!\cdots\!76\)\( \beta_{1}) q^{48} +(\)\(10\!\cdots\!97\)\( - \)\(19\!\cdots\!40\)\( \beta_{1} - \)\(10\!\cdots\!56\)\( \beta_{2} - \)\(70\!\cdots\!64\)\( \beta_{3}) q^{49} +(\)\(20\!\cdots\!00\)\( - \)\(39\!\cdots\!80\)\( \beta_{1} + \)\(15\!\cdots\!80\)\( \beta_{2} - \)\(33\!\cdots\!00\)\( \beta_{3}) q^{50} +(-\)\(26\!\cdots\!28\)\( - \)\(42\!\cdots\!26\)\( \beta_{1} + \)\(22\!\cdots\!28\)\( \beta_{2} + \)\(70\!\cdots\!82\)\( \beta_{3}) q^{51} +(\)\(17\!\cdots\!04\)\( - \)\(86\!\cdots\!80\)\( \beta_{1} - \)\(15\!\cdots\!56\)\( \beta_{2} - \)\(68\!\cdots\!64\)\( \beta_{3}) q^{52} +(-\)\(31\!\cdots\!06\)\( + \)\(22\!\cdots\!07\)\( \beta_{1} - \)\(25\!\cdots\!87\)\( \beta_{2} - \)\(26\!\cdots\!28\)\( \beta_{3}) q^{53} +(-\)\(11\!\cdots\!20\)\( - \)\(29\!\cdots\!00\)\( \beta_{1} - \)\(31\!\cdots\!36\)\( \beta_{2} + \)\(61\!\cdots\!16\)\( \beta_{3}) q^{54} +(-\)\(10\!\cdots\!60\)\( + \)\(15\!\cdots\!28\)\( \beta_{1} + \)\(20\!\cdots\!62\)\( \beta_{2} - \)\(52\!\cdots\!25\)\( \beta_{3}) q^{55} +(\)\(61\!\cdots\!48\)\( - \)\(86\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!04\)\( \beta_{2} - \)\(13\!\cdots\!76\)\( \beta_{3}) q^{56} +(-\)\(27\!\cdots\!80\)\( - \)\(19\!\cdots\!70\)\( \beta_{1} + \)\(34\!\cdots\!94\)\( \beta_{2} + \)\(32\!\cdots\!36\)\( \beta_{3}) q^{57} +(\)\(24\!\cdots\!00\)\( - \)\(98\!\cdots\!20\)\( \beta_{1} + \)\(11\!\cdots\!72\)\( \beta_{2} - \)\(45\!\cdots\!32\)\( \beta_{3}) q^{58} +(\)\(12\!\cdots\!00\)\( + \)\(79\!\cdots\!45\)\( \beta_{1} - \)\(20\!\cdots\!36\)\( \beta_{2} - \)\(65\!\cdots\!84\)\( \beta_{3}) q^{59} +(-\)\(16\!\cdots\!20\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} + \)\(10\!\cdots\!04\)\( \beta_{2} + \)\(28\!\cdots\!00\)\( \beta_{3}) q^{60} +(\)\(34\!\cdots\!02\)\( + \)\(39\!\cdots\!71\)\( \beta_{1} + \)\(57\!\cdots\!41\)\( \beta_{2} - \)\(44\!\cdots\!96\)\( \beta_{3}) q^{61} +(\)\(12\!\cdots\!52\)\( - \)\(72\!\cdots\!92\)\( \beta_{1} + \)\(21\!\cdots\!28\)\( \beta_{2} - \)\(39\!\cdots\!68\)\( \beta_{3}) q^{62} +(\)\(63\!\cdots\!04\)\( + \)\(31\!\cdots\!08\)\( \beta_{1} + \)\(44\!\cdots\!06\)\( \beta_{2} - \)\(66\!\cdots\!61\)\( \beta_{3}) q^{63} +\)\(17\!\cdots\!76\)\( q^{64} +(-\)\(70\!\cdots\!20\)\( + \)\(59\!\cdots\!16\)\( \beta_{1} + \)\(15\!\cdots\!64\)\( \beta_{2} + \)\(66\!\cdots\!00\)\( \beta_{3}) q^{65} +(-\)\(19\!\cdots\!72\)\( - \)\(14\!\cdots\!60\)\( \beta_{1} - \)\(16\!\cdots\!52\)\( \beta_{2} - \)\(17\!\cdots\!88\)\( \beta_{3}) q^{66} +(-\)\(34\!\cdots\!12\)\( - \)\(25\!\cdots\!25\)\( \beta_{1} + \)\(32\!\cdots\!72\)\( \beta_{2} - \)\(19\!\cdots\!82\)\( \beta_{3}) q^{67} +(\)\(51\!\cdots\!28\)\( + \)\(45\!\cdots\!56\)\( \beta_{1} + \)\(36\!\cdots\!68\)\( \beta_{2} - \)\(50\!\cdots\!08\)\( \beta_{3}) q^{68} +(-\)\(16\!\cdots\!44\)\( - \)\(47\!\cdots\!40\)\( \beta_{1} + \)\(35\!\cdots\!56\)\( \beta_{2} + \)\(36\!\cdots\!64\)\( \beta_{3}) q^{69} +(-\)\(30\!\cdots\!40\)\( + \)\(15\!\cdots\!52\)\( \beta_{1} + \)\(31\!\cdots\!08\)\( \beta_{2} + \)\(96\!\cdots\!00\)\( \beta_{3}) q^{70} +(\)\(13\!\cdots\!12\)\( + \)\(50\!\cdots\!32\)\( \beta_{1} - \)\(45\!\cdots\!62\)\( \beta_{2} - \)\(13\!\cdots\!53\)\( \beta_{3}) q^{71} +(\)\(33\!\cdots\!48\)\( + \)\(20\!\cdots\!68\)\( \beta_{1} + \)\(16\!\cdots\!92\)\( \beta_{2} - \)\(22\!\cdots\!52\)\( \beta_{3}) q^{72} +(\)\(12\!\cdots\!14\)\( - \)\(83\!\cdots\!58\)\( \beta_{1} + \)\(77\!\cdots\!18\)\( \beta_{2} + \)\(17\!\cdots\!92\)\( \beta_{3}) q^{73} +(\)\(78\!\cdots\!48\)\( + \)\(95\!\cdots\!76\)\( \beta_{1} - \)\(11\!\cdots\!32\)\( \beta_{2} - \)\(42\!\cdots\!08\)\( \beta_{3}) q^{74} +(\)\(21\!\cdots\!00\)\( - \)\(29\!\cdots\!95\)\( \beta_{1} + \)\(15\!\cdots\!20\)\( \beta_{2} + \)\(23\!\cdots\!00\)\( \beta_{3}) q^{75} +(\)\(17\!\cdots\!80\)\( + \)\(42\!\cdots\!28\)\( \beta_{1} + \)\(18\!\cdots\!12\)\( \beta_{2} + \)\(19\!\cdots\!28\)\( \beta_{3}) q^{76} +(\)\(27\!\cdots\!56\)\( - \)\(16\!\cdots\!04\)\( \beta_{1} - \)\(37\!\cdots\!72\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3}) q^{77} +(\)\(52\!\cdots\!96\)\( - \)\(57\!\cdots\!12\)\( \beta_{1} - \)\(77\!\cdots\!00\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3}) q^{78} +(-\)\(12\!\cdots\!60\)\( - \)\(56\!\cdots\!16\)\( \beta_{1} - \)\(14\!\cdots\!96\)\( \beta_{2} + \)\(32\!\cdots\!26\)\( \beta_{3}) q^{79} +(\)\(91\!\cdots\!20\)\( + \)\(26\!\cdots\!44\)\( \beta_{1} + \)\(14\!\cdots\!76\)\( \beta_{2}) q^{80} +(\)\(94\!\cdots\!01\)\( + \)\(14\!\cdots\!58\)\( \beta_{1} + \)\(24\!\cdots\!14\)\( \beta_{2} - \)\(45\!\cdots\!84\)\( \beta_{3}) q^{81} +(\)\(66\!\cdots\!72\)\( - \)\(38\!\cdots\!96\)\( \beta_{1} + \)\(70\!\cdots\!76\)\( \beta_{2} - \)\(38\!\cdots\!56\)\( \beta_{3}) q^{82} +(-\)\(59\!\cdots\!16\)\( - \)\(11\!\cdots\!81\)\( \beta_{1} - \)\(17\!\cdots\!32\)\( \beta_{2} + \)\(81\!\cdots\!92\)\( \beta_{3}) q^{83} +(-\)\(44\!\cdots\!48\)\( - \)\(14\!\cdots\!92\)\( \beta_{1} - \)\(17\!\cdots\!72\)\( \beta_{2} + \)\(71\!\cdots\!32\)\( \beta_{3}) q^{84} +(\)\(17\!\cdots\!60\)\( - \)\(11\!\cdots\!58\)\( \beta_{1} + \)\(29\!\cdots\!18\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3}) q^{85} +(\)\(10\!\cdots\!04\)\( - \)\(29\!\cdots\!40\)\( \beta_{1} - \)\(71\!\cdots\!64\)\( \beta_{2} + \)\(50\!\cdots\!84\)\( \beta_{3}) q^{86} +(\)\(62\!\cdots\!00\)\( + \)\(51\!\cdots\!56\)\( \beta_{1} + \)\(65\!\cdots\!10\)\( \beta_{2} - \)\(11\!\cdots\!35\)\( \beta_{3}) q^{87} +(\)\(10\!\cdots\!72\)\( + \)\(30\!\cdots\!96\)\( \beta_{1} - \)\(46\!\cdots\!52\)\( \beta_{2} - \)\(81\!\cdots\!88\)\( \beta_{3}) q^{88} +(\)\(68\!\cdots\!90\)\( - \)\(78\!\cdots\!90\)\( \beta_{1} - \)\(38\!\cdots\!50\)\( \beta_{2} + \)\(38\!\cdots\!00\)\( \beta_{3}) q^{89} +(\)\(11\!\cdots\!60\)\( + \)\(46\!\cdots\!72\)\( \beta_{1} + \)\(22\!\cdots\!88\)\( \beta_{2} - \)\(21\!\cdots\!00\)\( \beta_{3}) q^{90} +(\)\(20\!\cdots\!92\)\( - \)\(13\!\cdots\!92\)\( \beta_{1} - \)\(68\!\cdots\!36\)\( \beta_{2} - \)\(22\!\cdots\!84\)\( \beta_{3}) q^{91} +(\)\(59\!\cdots\!44\)\( + \)\(27\!\cdots\!12\)\( \beta_{1} + \)\(61\!\cdots\!36\)\( \beta_{2} + \)\(51\!\cdots\!84\)\( \beta_{3}) q^{92} +(\)\(43\!\cdots\!48\)\( - \)\(72\!\cdots\!04\)\( \beta_{1} + \)\(54\!\cdots\!44\)\( \beta_{2} - \)\(59\!\cdots\!64\)\( \beta_{3}) q^{93} +(\)\(18\!\cdots\!88\)\( + \)\(49\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!32\)\( \beta_{2} + \)\(51\!\cdots\!08\)\( \beta_{3}) q^{94} +(\)\(95\!\cdots\!00\)\( - \)\(23\!\cdots\!20\)\( \beta_{1} + \)\(26\!\cdots\!70\)\( \beta_{2} - \)\(20\!\cdots\!75\)\( \beta_{3}) q^{95} +(-\)\(28\!\cdots\!76\)\( - \)\(16\!\cdots\!76\)\( \beta_{1}) q^{96} +(-\)\(38\!\cdots\!02\)\( - \)\(24\!\cdots\!54\)\( \beta_{1} - \)\(67\!\cdots\!18\)\( \beta_{2} + \)\(13\!\cdots\!08\)\( \beta_{3}) q^{97} +(\)\(11\!\cdots\!72\)\( - \)\(21\!\cdots\!40\)\( \beta_{1} - \)\(11\!\cdots\!56\)\( \beta_{2} - \)\(77\!\cdots\!64\)\( \beta_{3}) q^{98} +(\)\(58\!\cdots\!56\)\( + \)\(37\!\cdots\!73\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} - \)\(25\!\cdots\!36\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4398046511104q^{2} - 7111737378636958704q^{3} + \)\(48\!\cdots\!04\)\(q^{4} + \)\(24\!\cdots\!80\)\(q^{5} - \)\(78\!\cdots\!04\)\(q^{6} + \)\(18\!\cdots\!92\)\(q^{7} + \)\(53\!\cdots\!04\)\(q^{8} + \)\(10\!\cdots\!92\)\(q^{9} + O(q^{10}) \) \( 4q + 4398046511104q^{2} - 7111737378636958704q^{3} + \)\(48\!\cdots\!04\)\(q^{4} + \)\(24\!\cdots\!80\)\(q^{5} - \)\(78\!\cdots\!04\)\(q^{6} + \)\(18\!\cdots\!92\)\(q^{7} + \)\(53\!\cdots\!04\)\(q^{8} + \)\(10\!\cdots\!92\)\(q^{9} + \)\(27\!\cdots\!80\)\(q^{10} + \)\(32\!\cdots\!88\)\(q^{11} - \)\(85\!\cdots\!04\)\(q^{12} + \)\(56\!\cdots\!16\)\(q^{13} + \)\(20\!\cdots\!92\)\(q^{14} - \)\(54\!\cdots\!80\)\(q^{15} + \)\(58\!\cdots\!04\)\(q^{16} + \)\(17\!\cdots\!12\)\(q^{17} + \)\(11\!\cdots\!92\)\(q^{18} + \)\(57\!\cdots\!20\)\(q^{19} + \)\(30\!\cdots\!80\)\(q^{20} - \)\(14\!\cdots\!92\)\(q^{21} + \)\(36\!\cdots\!88\)\(q^{22} + \)\(19\!\cdots\!76\)\(q^{23} - \)\(94\!\cdots\!04\)\(q^{24} + \)\(73\!\cdots\!00\)\(q^{25} + \)\(62\!\cdots\!16\)\(q^{26} - \)\(40\!\cdots\!80\)\(q^{27} + \)\(22\!\cdots\!92\)\(q^{28} + \)\(89\!\cdots\!00\)\(q^{29} - \)\(60\!\cdots\!80\)\(q^{30} + \)\(44\!\cdots\!08\)\(q^{31} + \)\(64\!\cdots\!04\)\(q^{32} - \)\(69\!\cdots\!88\)\(q^{33} + \)\(18\!\cdots\!12\)\(q^{34} - \)\(11\!\cdots\!60\)\(q^{35} + \)\(12\!\cdots\!92\)\(q^{36} + \)\(28\!\cdots\!92\)\(q^{37} + \)\(63\!\cdots\!20\)\(q^{38} + \)\(18\!\cdots\!84\)\(q^{39} + \)\(33\!\cdots\!80\)\(q^{40} + \)\(24\!\cdots\!88\)\(q^{41} - \)\(16\!\cdots\!92\)\(q^{42} + \)\(39\!\cdots\!16\)\(q^{43} + \)\(39\!\cdots\!88\)\(q^{44} + \)\(40\!\cdots\!40\)\(q^{45} + \)\(21\!\cdots\!76\)\(q^{46} + \)\(68\!\cdots\!52\)\(q^{47} - \)\(10\!\cdots\!04\)\(q^{48} + \)\(40\!\cdots\!88\)\(q^{49} + \)\(80\!\cdots\!00\)\(q^{50} - \)\(10\!\cdots\!12\)\(q^{51} + \)\(68\!\cdots\!16\)\(q^{52} - \)\(12\!\cdots\!24\)\(q^{53} - \)\(44\!\cdots\!80\)\(q^{54} - \)\(42\!\cdots\!40\)\(q^{55} + \)\(24\!\cdots\!92\)\(q^{56} - \)\(10\!\cdots\!20\)\(q^{57} + \)\(98\!\cdots\!00\)\(q^{58} + \)\(48\!\cdots\!00\)\(q^{59} - \)\(66\!\cdots\!80\)\(q^{60} + \)\(13\!\cdots\!08\)\(q^{61} + \)\(48\!\cdots\!08\)\(q^{62} + \)\(25\!\cdots\!16\)\(q^{63} + \)\(70\!\cdots\!04\)\(q^{64} - \)\(28\!\cdots\!80\)\(q^{65} - \)\(76\!\cdots\!88\)\(q^{66} - \)\(13\!\cdots\!48\)\(q^{67} + \)\(20\!\cdots\!12\)\(q^{68} - \)\(67\!\cdots\!76\)\(q^{69} - \)\(12\!\cdots\!60\)\(q^{70} + \)\(53\!\cdots\!48\)\(q^{71} + \)\(13\!\cdots\!92\)\(q^{72} + \)\(48\!\cdots\!56\)\(q^{73} + \)\(31\!\cdots\!92\)\(q^{74} + \)\(86\!\cdots\!00\)\(q^{75} + \)\(69\!\cdots\!20\)\(q^{76} + \)\(11\!\cdots\!24\)\(q^{77} + \)\(20\!\cdots\!84\)\(q^{78} - \)\(50\!\cdots\!40\)\(q^{79} + \)\(36\!\cdots\!80\)\(q^{80} + \)\(37\!\cdots\!04\)\(q^{81} + \)\(26\!\cdots\!88\)\(q^{82} - \)\(23\!\cdots\!64\)\(q^{83} - \)\(17\!\cdots\!92\)\(q^{84} + \)\(68\!\cdots\!40\)\(q^{85} + \)\(42\!\cdots\!16\)\(q^{86} + \)\(24\!\cdots\!00\)\(q^{87} + \)\(43\!\cdots\!88\)\(q^{88} + \)\(27\!\cdots\!60\)\(q^{89} + \)\(44\!\cdots\!40\)\(q^{90} + \)\(83\!\cdots\!68\)\(q^{91} + \)\(23\!\cdots\!76\)\(q^{92} + \)\(17\!\cdots\!92\)\(q^{93} + \)\(74\!\cdots\!52\)\(q^{94} + \)\(38\!\cdots\!00\)\(q^{95} - \)\(11\!\cdots\!04\)\(q^{96} - \)\(15\!\cdots\!08\)\(q^{97} + \)\(44\!\cdots\!88\)\(q^{98} + \)\(23\!\cdots\!24\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 376262508638451474994292856206343 x^{2} - 1502772308966385407243870196944706973427347215700 x + 15227343706090890463549797596060496062977217468321124839077527200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 1920 \nu - 960 \)
\(\beta_{2}\)\(=\)\((\)\(-1897922560 \nu^{3} + 27683303009004195755745280 \nu^{2} + 394650807354726094215039393568006977804160 \nu - 3068985417985209724798726201550810822959831937964908616640\)\()/ \)\(35\!\cdots\!33\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-1419635992821760 \nu^{3} + 12867168353563916791387697889280 \nu^{2} + 342164290928400366156726594216579624370054435840 \nu - 820674277774218300967859756654645837418612775903039903512455680\)\()/ \)\(35\!\cdots\!33\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 960\)\()/1920\)
\(\nu^{2}\)\(=\)\((\)\(-8326 \beta_{3} + 6227803771 \beta_{2} + 5751282879037881593 \beta_{1} + 346763527961196879354740296279767552000\)\()/1843200\)
\(\nu^{3}\)\(=\)\((\)\(-15544823466970129452689 \beta_{3} + 7225216604784425932663990514 \beta_{2} + 36289229870635194014447654419165125982 \beta_{1} + 265911352309136925089857285612416125988717264676415078400\)\()/ 235929600 \)

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
2.03307e16
4.77338e15
−1.17482e16
−1.33560e16
1.09951e12 −4.08130e19 1.20893e24 2.59550e28 −4.48743e31 1.64669e34 1.32923e36 1.22227e39 2.85378e40
1.2 1.09951e12 −1.09428e19 1.20893e24 −1.35941e28 −1.20318e31 −2.01060e34 1.32923e36 −3.23681e38 −1.49469e40
1.3 1.09951e12 2.07785e19 1.20893e24 −2.06533e28 2.28462e31 2.86366e34 1.32923e36 −1.16794e37 −2.27085e40
1.4 1.09951e12 2.38655e19 1.20893e24 3.32056e28 2.62404e31 −6.60461e33 1.32923e36 1.26136e38 3.65099e40
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3}^{4} + \)\(71\!\cdots\!04\)\( T_{3}^{3} - \)\(13\!\cdots\!44\)\( T_{3}^{2} + \)\(57\!\cdots\!04\)\( T_{3} + \)\(22\!\cdots\!76\)\( \) acting on \(S_{82}^{\mathrm{new}}(\Gamma_0(2))\).