Properties

Label 2.82.a.a
Level 2
Weight 82
Character orbit 2.a
Self dual Yes
Analytic conductor 83.100
Analytic rank 1
Dimension 3
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{10}\cdot 5^{3}\cdot 7 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(-1099511627776 q^{2}\) \(+(4201453557463404996 + \beta_{1}) q^{3}\) \(+\)\(12\!\cdots\!76\)\( q^{4}\) \(+(-\)\(69\!\cdots\!50\)\( + 298982601 \beta_{1} + 769 \beta_{2}) q^{5}\) \(+(-\)\(46\!\cdots\!96\)\( - 1099511627776 \beta_{1}) q^{6}\) \(+(-\)\(18\!\cdots\!08\)\( - 446074228826442 \beta_{1} + 572763764 \beta_{2}) q^{7}\) \(-\)\(13\!\cdots\!76\)\( q^{8}\) \(+(-\)\(39\!\cdots\!87\)\( + 4707282525607846350 \beta_{1} + 9842345463582 \beta_{2}) q^{9}\) \(+O(q^{10})\) \( q\) \(-1099511627776 q^{2}\) \(+(4201453557463404996 + \beta_{1}) q^{3}\) \(+\)\(12\!\cdots\!76\)\( q^{4}\) \(+(-\)\(69\!\cdots\!50\)\( + 298982601 \beta_{1} + 769 \beta_{2}) q^{5}\) \(+(-\)\(46\!\cdots\!96\)\( - 1099511627776 \beta_{1}) q^{6}\) \(+(-\)\(18\!\cdots\!08\)\( - 446074228826442 \beta_{1} + 572763764 \beta_{2}) q^{7}\) \(-\)\(13\!\cdots\!76\)\( q^{8}\) \(+(-\)\(39\!\cdots\!87\)\( + 4707282525607846350 \beta_{1} + 9842345463582 \beta_{2}) q^{9}\) \(+(\)\(76\!\cdots\!00\)\( - \)\(32\!\cdots\!76\)\( \beta_{1} - 845524441759744 \beta_{2}) q^{10}\) \(+(\)\(42\!\cdots\!12\)\( + \)\(24\!\cdots\!39\)\( \beta_{1} - 23868188313924152 \beta_{2}) q^{11}\) \(+(\)\(50\!\cdots\!96\)\( + \)\(12\!\cdots\!76\)\( \beta_{1}) q^{12}\) \(+(-\)\(67\!\cdots\!14\)\( + \)\(32\!\cdots\!61\)\( \beta_{1} - 34303936689160258679 \beta_{2}) q^{13}\) \(+(\)\(20\!\cdots\!08\)\( + \)\(49\!\cdots\!92\)\( \beta_{1} - \)\(62\!\cdots\!64\)\( \beta_{2}) q^{14}\) \(+(\)\(86\!\cdots\!00\)\( + \)\(71\!\cdots\!66\)\( \beta_{1} + \)\(90\!\cdots\!04\)\( \beta_{2}) q^{15}\) \(+\)\(14\!\cdots\!76\)\( q^{16}\) \(+(\)\(45\!\cdots\!82\)\( - \)\(71\!\cdots\!54\)\( \beta_{1} - \)\(29\!\cdots\!58\)\( \beta_{2}) q^{17}\) \(+(\)\(43\!\cdots\!12\)\( - \)\(51\!\cdots\!00\)\( \beta_{1} - \)\(10\!\cdots\!32\)\( \beta_{2}) q^{18}\) \(+(\)\(15\!\cdots\!20\)\( - \)\(20\!\cdots\!75\)\( \beta_{1} - \)\(19\!\cdots\!68\)\( \beta_{2}) q^{19}\) \(+(-\)\(84\!\cdots\!00\)\( + \)\(36\!\cdots\!76\)\( \beta_{1} + \)\(92\!\cdots\!44\)\( \beta_{2}) q^{20}\) \(+(-\)\(17\!\cdots\!68\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} + \)\(13\!\cdots\!88\)\( \beta_{2}) q^{21}\) \(+(-\)\(46\!\cdots\!12\)\( - \)\(26\!\cdots\!64\)\( \beta_{1} + \)\(26\!\cdots\!52\)\( \beta_{2}) q^{22}\) \(+(-\)\(10\!\cdots\!24\)\( - \)\(21\!\cdots\!18\)\( \beta_{1} + \)\(20\!\cdots\!44\)\( \beta_{2}) q^{23}\) \(+(-\)\(55\!\cdots\!96\)\( - \)\(13\!\cdots\!76\)\( \beta_{1}) q^{24}\) \(+(\)\(93\!\cdots\!75\)\( + \)\(79\!\cdots\!00\)\( \beta_{1} - \)\(18\!\cdots\!00\)\( \beta_{2}) q^{25}\) \(+(\)\(74\!\cdots\!64\)\( - \)\(35\!\cdots\!36\)\( \beta_{1} + \)\(37\!\cdots\!04\)\( \beta_{2}) q^{26}\) \(+(-\)\(20\!\cdots\!40\)\( - \)\(30\!\cdots\!54\)\( \beta_{1} + \)\(12\!\cdots\!16\)\( \beta_{2}) q^{27}\) \(+(-\)\(22\!\cdots\!08\)\( - \)\(53\!\cdots\!92\)\( \beta_{1} + \)\(69\!\cdots\!64\)\( \beta_{2}) q^{28}\) \(+(\)\(71\!\cdots\!30\)\( - \)\(48\!\cdots\!03\)\( \beta_{1} - \)\(34\!\cdots\!43\)\( \beta_{2}) q^{29}\) \(+(-\)\(94\!\cdots\!00\)\( - \)\(79\!\cdots\!16\)\( \beta_{1} - \)\(99\!\cdots\!04\)\( \beta_{2}) q^{30}\) \(+(-\)\(12\!\cdots\!68\)\( - \)\(19\!\cdots\!64\)\( \beta_{1} + \)\(37\!\cdots\!48\)\( \beta_{2}) q^{31}\) \(-\)\(16\!\cdots\!76\)\( q^{32}\) \(+(\)\(27\!\cdots\!52\)\( - \)\(74\!\cdots\!78\)\( \beta_{1} - \)\(16\!\cdots\!78\)\( \beta_{2}) q^{33}\) \(+(-\)\(50\!\cdots\!32\)\( + \)\(78\!\cdots\!04\)\( \beta_{1} + \)\(32\!\cdots\!08\)\( \beta_{2}) q^{34}\) \(+(\)\(26\!\cdots\!00\)\( + \)\(34\!\cdots\!32\)\( \beta_{1} - \)\(14\!\cdots\!92\)\( \beta_{2}) q^{35}\) \(+(-\)\(47\!\cdots\!12\)\( + \)\(56\!\cdots\!00\)\( \beta_{1} + \)\(11\!\cdots\!32\)\( \beta_{2}) q^{36}\) \(+(-\)\(32\!\cdots\!38\)\( + \)\(13\!\cdots\!21\)\( \beta_{1} - \)\(23\!\cdots\!03\)\( \beta_{2}) q^{37}\) \(+(-\)\(17\!\cdots\!20\)\( + \)\(22\!\cdots\!00\)\( \beta_{1} + \)\(21\!\cdots\!68\)\( \beta_{2}) q^{38}\) \(+(\)\(12\!\cdots\!56\)\( - \)\(67\!\cdots\!62\)\( \beta_{1} + \)\(45\!\cdots\!00\)\( \beta_{2}) q^{39}\) \(+(\)\(92\!\cdots\!00\)\( - \)\(39\!\cdots\!76\)\( \beta_{1} - \)\(10\!\cdots\!44\)\( \beta_{2}) q^{40}\) \(+(\)\(10\!\cdots\!42\)\( - \)\(75\!\cdots\!00\)\( \beta_{1} - \)\(31\!\cdots\!04\)\( \beta_{2}) q^{41}\) \(+(\)\(19\!\cdots\!68\)\( - \)\(11\!\cdots\!96\)\( \beta_{1} - \)\(14\!\cdots\!88\)\( \beta_{2}) q^{42}\) \(+(\)\(30\!\cdots\!56\)\( + \)\(28\!\cdots\!99\)\( \beta_{1} + \)\(53\!\cdots\!04\)\( \beta_{2}) q^{43}\) \(+(\)\(51\!\cdots\!12\)\( + \)\(29\!\cdots\!64\)\( \beta_{1} - \)\(28\!\cdots\!52\)\( \beta_{2}) q^{44}\) \(+(\)\(62\!\cdots\!50\)\( + \)\(12\!\cdots\!53\)\( \beta_{1} - \)\(19\!\cdots\!43\)\( \beta_{2}) q^{45}\) \(+(\)\(11\!\cdots\!24\)\( + \)\(24\!\cdots\!68\)\( \beta_{1} - \)\(22\!\cdots\!44\)\( \beta_{2}) q^{46}\) \(+(\)\(54\!\cdots\!52\)\( + \)\(80\!\cdots\!64\)\( \beta_{1} + \)\(14\!\cdots\!88\)\( \beta_{2}) q^{47}\) \(+(\)\(61\!\cdots\!96\)\( + \)\(14\!\cdots\!76\)\( \beta_{1}) q^{48}\) \(+(\)\(28\!\cdots\!57\)\( - \)\(76\!\cdots\!16\)\( \beta_{1} - \)\(61\!\cdots\!76\)\( \beta_{2}) q^{49}\) \(+(-\)\(10\!\cdots\!00\)\( - \)\(87\!\cdots\!00\)\( \beta_{1} + \)\(20\!\cdots\!00\)\( \beta_{2}) q^{50}\) \(+(-\)\(25\!\cdots\!28\)\( - \)\(50\!\cdots\!18\)\( \beta_{1} - \)\(30\!\cdots\!32\)\( \beta_{2}) q^{51}\) \(+(-\)\(81\!\cdots\!64\)\( + \)\(38\!\cdots\!36\)\( \beta_{1} - \)\(41\!\cdots\!04\)\( \beta_{2}) q^{52}\) \(+(-\)\(30\!\cdots\!54\)\( + \)\(14\!\cdots\!25\)\( \beta_{1} + \)\(12\!\cdots\!97\)\( \beta_{2}) q^{53}\) \(+(\)\(22\!\cdots\!40\)\( + \)\(33\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!16\)\( \beta_{2}) q^{54}\) \(+(-\)\(15\!\cdots\!00\)\( - \)\(11\!\cdots\!98\)\( \beta_{1} + \)\(81\!\cdots\!88\)\( \beta_{2}) q^{55}\) \(+(\)\(24\!\cdots\!08\)\( + \)\(59\!\cdots\!92\)\( \beta_{1} - \)\(76\!\cdots\!64\)\( \beta_{2}) q^{56}\) \(+(-\)\(72\!\cdots\!80\)\( - \)\(21\!\cdots\!94\)\( \beta_{1} - \)\(35\!\cdots\!34\)\( \beta_{2}) q^{57}\) \(+(-\)\(78\!\cdots\!80\)\( + \)\(53\!\cdots\!28\)\( \beta_{1} + \)\(38\!\cdots\!68\)\( \beta_{2}) q^{58}\) \(+(\)\(57\!\cdots\!60\)\( - \)\(29\!\cdots\!61\)\( \beta_{1} + \)\(15\!\cdots\!44\)\( \beta_{2}) q^{59}\) \(+(\)\(10\!\cdots\!00\)\( + \)\(86\!\cdots\!16\)\( \beta_{1} + \)\(10\!\cdots\!04\)\( \beta_{2}) q^{60}\) \(+(\)\(12\!\cdots\!62\)\( - \)\(39\!\cdots\!23\)\( \beta_{1} + \)\(10\!\cdots\!41\)\( \beta_{2}) q^{61}\) \(+(\)\(14\!\cdots\!68\)\( + \)\(21\!\cdots\!64\)\( \beta_{1} - \)\(41\!\cdots\!48\)\( \beta_{2}) q^{62}\) \(+(\)\(32\!\cdots\!96\)\( + \)\(32\!\cdots\!66\)\( \beta_{1} - \)\(15\!\cdots\!76\)\( \beta_{2}) q^{63}\) \(+\)\(17\!\cdots\!76\)\( q^{64}\) \(+(-\)\(14\!\cdots\!00\)\( - \)\(19\!\cdots\!24\)\( \beta_{1} + \)\(81\!\cdots\!44\)\( \beta_{2}) q^{65}\) \(+(-\)\(29\!\cdots\!52\)\( + \)\(81\!\cdots\!28\)\( \beta_{1} + \)\(18\!\cdots\!28\)\( \beta_{2}) q^{66}\) \(+(-\)\(15\!\cdots\!68\)\( - \)\(99\!\cdots\!47\)\( \beta_{1} - \)\(90\!\cdots\!52\)\( \beta_{2}) q^{67}\) \(+(\)\(55\!\cdots\!32\)\( - \)\(86\!\cdots\!04\)\( \beta_{1} - \)\(36\!\cdots\!08\)\( \beta_{2}) q^{68}\) \(+(-\)\(12\!\cdots\!04\)\( - \)\(63\!\cdots\!84\)\( \beta_{1} - \)\(52\!\cdots\!04\)\( \beta_{2}) q^{69}\) \(+(-\)\(29\!\cdots\!00\)\( - \)\(37\!\cdots\!32\)\( \beta_{1} + \)\(15\!\cdots\!92\)\( \beta_{2}) q^{70}\) \(+(-\)\(22\!\cdots\!28\)\( + \)\(37\!\cdots\!70\)\( \beta_{1} - \)\(72\!\cdots\!92\)\( \beta_{2}) q^{71}\) \(+(\)\(52\!\cdots\!12\)\( - \)\(62\!\cdots\!00\)\( \beta_{1} - \)\(13\!\cdots\!32\)\( \beta_{2}) q^{72}\) \(+(\)\(95\!\cdots\!26\)\( + \)\(15\!\cdots\!90\)\( \beta_{1} + \)\(14\!\cdots\!62\)\( \beta_{2}) q^{73}\) \(+(\)\(35\!\cdots\!88\)\( - \)\(15\!\cdots\!96\)\( \beta_{1} + \)\(25\!\cdots\!28\)\( \beta_{2}) q^{74}\) \(+(\)\(34\!\cdots\!00\)\( - \)\(24\!\cdots\!25\)\( \beta_{1} - \)\(71\!\cdots\!00\)\( \beta_{2}) q^{75}\) \(+(\)\(19\!\cdots\!20\)\( - \)\(24\!\cdots\!00\)\( \beta_{1} - \)\(24\!\cdots\!68\)\( \beta_{2}) q^{76}\) \(+(-\)\(10\!\cdots\!96\)\( - \)\(61\!\cdots\!44\)\( \beta_{1} + \)\(53\!\cdots\!72\)\( \beta_{2}) q^{77}\) \(+(-\)\(13\!\cdots\!56\)\( + \)\(74\!\cdots\!12\)\( \beta_{1} - \)\(49\!\cdots\!00\)\( \beta_{2}) q^{78}\) \(+(-\)\(53\!\cdots\!20\)\( + \)\(22\!\cdots\!88\)\( \beta_{1} + \)\(31\!\cdots\!64\)\( \beta_{2}) q^{79}\) \(+(-\)\(10\!\cdots\!00\)\( + \)\(43\!\cdots\!76\)\( \beta_{1} + \)\(11\!\cdots\!44\)\( \beta_{2}) q^{80}\) \(+(-\)\(99\!\cdots\!79\)\( - \)\(18\!\cdots\!38\)\( \beta_{1} - \)\(63\!\cdots\!66\)\( \beta_{2}) q^{81}\) \(+(-\)\(11\!\cdots\!92\)\( + \)\(83\!\cdots\!00\)\( \beta_{1} + \)\(34\!\cdots\!04\)\( \beta_{2}) q^{82}\) \(+(-\)\(51\!\cdots\!84\)\( - \)\(81\!\cdots\!47\)\( \beta_{1} - \)\(20\!\cdots\!28\)\( \beta_{2}) q^{83}\) \(+(-\)\(20\!\cdots\!68\)\( + \)\(12\!\cdots\!96\)\( \beta_{1} + \)\(16\!\cdots\!88\)\( \beta_{2}) q^{84}\) \(+(-\)\(17\!\cdots\!00\)\( - \)\(34\!\cdots\!58\)\( \beta_{1} + \)\(58\!\cdots\!98\)\( \beta_{2}) q^{85}\) \(+(-\)\(33\!\cdots\!56\)\( - \)\(31\!\cdots\!24\)\( \beta_{1} - \)\(58\!\cdots\!04\)\( \beta_{2}) q^{86}\) \(+(-\)\(18\!\cdots\!20\)\( - \)\(58\!\cdots\!46\)\( \beta_{1} - \)\(75\!\cdots\!80\)\( \beta_{2}) q^{87}\) \(+(-\)\(56\!\cdots\!12\)\( - \)\(31\!\cdots\!64\)\( \beta_{1} + \)\(31\!\cdots\!52\)\( \beta_{2}) q^{88}\) \(+(-\)\(28\!\cdots\!10\)\( + \)\(42\!\cdots\!70\)\( \beta_{1} - \)\(54\!\cdots\!10\)\( \beta_{2}) q^{89}\) \(+(-\)\(68\!\cdots\!00\)\( - \)\(13\!\cdots\!28\)\( \beta_{1} + \)\(21\!\cdots\!68\)\( \beta_{2}) q^{90}\) \(+(-\)\(19\!\cdots\!88\)\( + \)\(55\!\cdots\!32\)\( \beta_{1} + \)\(30\!\cdots\!64\)\( \beta_{2}) q^{91}\) \(+(-\)\(12\!\cdots\!24\)\( - \)\(26\!\cdots\!68\)\( \beta_{1} + \)\(25\!\cdots\!44\)\( \beta_{2}) q^{92}\) \(+(-\)\(80\!\cdots\!28\)\( - \)\(70\!\cdots\!20\)\( \beta_{1} - \)\(16\!\cdots\!24\)\( \beta_{2}) q^{93}\) \(+(-\)\(59\!\cdots\!52\)\( - \)\(88\!\cdots\!64\)\( \beta_{1} - \)\(15\!\cdots\!88\)\( \beta_{2}) q^{94}\) \(+(-\)\(14\!\cdots\!00\)\( - \)\(28\!\cdots\!90\)\( \beta_{1} + \)\(40\!\cdots\!40\)\( \beta_{2}) q^{95}\) \(+(-\)\(67\!\cdots\!96\)\( - \)\(16\!\cdots\!76\)\( \beta_{1}) q^{96}\) \(+(-\)\(17\!\cdots\!98\)\( + \)\(59\!\cdots\!62\)\( \beta_{1} - \)\(39\!\cdots\!02\)\( \beta_{2}) q^{97}\) \(+(-\)\(31\!\cdots\!32\)\( + \)\(84\!\cdots\!16\)\( \beta_{1} + \)\(67\!\cdots\!76\)\( \beta_{2}) q^{98}\) \(+(-\)\(18\!\cdots\!44\)\( - \)\(13\!\cdots\!21\)\( \beta_{1} + \)\(92\!\cdots\!96\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 3298534883328q^{2} \) \(\mathstrut +\mathstrut 12604360672390214988q^{3} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!28\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!88\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!24\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!28\)\(q^{8} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!61\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 3298534883328q^{2} \) \(\mathstrut +\mathstrut 12604360672390214988q^{3} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!28\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!50\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!88\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!24\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!28\)\(q^{8} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!61\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!36\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!88\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!42\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!24\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!28\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!46\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!36\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!60\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!04\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!36\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!72\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!88\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!25\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!92\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!20\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!24\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!90\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!04\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!28\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!56\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!96\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!36\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!14\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!60\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!68\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!26\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!04\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!68\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!36\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!50\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!72\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!56\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!88\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!71\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!84\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!92\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!62\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!20\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!24\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!40\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!40\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!80\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!86\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!04\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!88\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!28\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!56\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!04\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!96\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!12\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!84\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!36\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!78\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!64\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!60\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!88\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!68\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!60\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!37\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!76\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!52\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!04\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!68\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!60\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!36\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!30\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!64\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!72\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!84\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!56\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!88\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!94\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!96\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!32\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(215736607049989005852988854472\) \(x\mathstrut +\mathstrut \) \(10253113298782277175624314846636353390456960\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 51840 \nu - 17280 \)
\(\beta_{2}\)\(=\)\((\)\( 204800 \nu^{2} + 14599998377556263040 \nu - 29455238082558503799127537449399680 \)\()/\)\(750064431\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(17280\)\()/51840\)
\(\nu^{2}\)\(=\)\((\)\(20251739637\) \(\beta_{2}\mathstrut -\mathstrut \) \(7604165821643887\) \(\beta_{1}\mathstrut +\mathstrut \) \(795291428229079471176458113127424000\)\()/5529600\)

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
−4.86628e14
4.80400e13
4.38588e14
−1.09951e12 −2.10253e19 1.20893e24 −2.29740e27 2.31176e31 2.01848e34 −1.32923e36 −1.36253e36 2.52602e39
1.2 −1.09951e12 6.69185e18 1.20893e24 −3.52449e28 −7.35776e30 −2.28912e34 −1.32923e36 −3.98646e38 3.87522e40
1.3 −1.09951e12 2.69378e19 1.20893e24 1.65593e28 −2.96185e31 2.15367e33 −1.32923e36 2.82220e38 −1.82071e40
\(n\): 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 \(T_{3}^{3} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!88\)\( T_{3}^{2} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!52\)\( T_{3} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!64\)\( \) acting on \(S_{82}^{\mathrm{new}}(\Gamma_0(2))\).