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

Related objects

Downloads

Learn more about

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.

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

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

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

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 \(T_{3}^{4} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!04\)\( T_{3}^{3} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!44\)\( T_{3}^{2} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!04\)\( T_{3} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!76\)\( \) acting on \(S_{82}^{\mathrm{new}}(\Gamma_0(2))\).