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.

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

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

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

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 \( T_{3}^{3} - \)\(12\!\cdots\!88\)\( T_{3}^{2} - \)\(52\!\cdots\!52\)\( T_{3} + \)\(37\!\cdots\!64\)\( \) acting on \(S_{82}^{\mathrm{new}}(\Gamma_0(2))\).