Properties

Label 2.84.a.b
Level 2
Weight 84
Character orbit 2.a
Self dual Yes
Analytic conductor 87.254
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 84 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(87.2544256533\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{11}\cdot 5^{2}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q +2199023255552 q^{2} +(-5621858992801999908 - \beta_{1}) q^{3} +\)\(48\!\cdots\!04\)\( q^{4} +(-\)\(29\!\cdots\!90\)\( + 138290647 \beta_{1} - 1657 \beta_{2}) q^{5} +(-\)\(12\!\cdots\!16\)\( - 2199023255552 \beta_{1}) q^{6} +(\)\(24\!\cdots\!96\)\( - 274375425188998 \beta_{1} - 1240461740 \beta_{2}) q^{7} +\)\(10\!\cdots\!08\)\( q^{8} +(\)\(96\!\cdots\!37\)\( - 14493938576442927174 \beta_{1} + 56303458286490 \beta_{2}) q^{9} +O(q^{10})\) \( q +2199023255552 q^{2} +(-5621858992801999908 - \beta_{1}) q^{3} +\)\(48\!\cdots\!04\)\( q^{4} +(-\)\(29\!\cdots\!90\)\( + 138290647 \beta_{1} - 1657 \beta_{2}) q^{5} +(-\)\(12\!\cdots\!16\)\( - 2199023255552 \beta_{1}) q^{6} +(\)\(24\!\cdots\!96\)\( - 274375425188998 \beta_{1} - 1240461740 \beta_{2}) q^{7} +\)\(10\!\cdots\!08\)\( q^{8} +(\)\(96\!\cdots\!37\)\( - 14493938576442927174 \beta_{1} + 56303458286490 \beta_{2}) q^{9} +(-\)\(65\!\cdots\!80\)\( + \)\(30\!\cdots\!44\)\( \beta_{1} - 3643781534449664 \beta_{2}) q^{10} +(-\)\(83\!\cdots\!88\)\( + \)\(13\!\cdots\!49\)\( \beta_{1} - 78240138957300760 \beta_{2}) q^{11} +(-\)\(27\!\cdots\!32\)\( - \)\(48\!\cdots\!04\)\( \beta_{1}) q^{12} +(-\)\(44\!\cdots\!58\)\( + \)\(16\!\cdots\!23\)\( \beta_{1} + \)\(36\!\cdots\!35\)\( \beta_{2}) q^{13} +(\)\(53\!\cdots\!92\)\( - \)\(60\!\cdots\!96\)\( \beta_{1} - \)\(27\!\cdots\!80\)\( \beta_{2}) q^{14} +(-\)\(51\!\cdots\!80\)\( + \)\(85\!\cdots\!74\)\( \beta_{1} + \)\(44\!\cdots\!56\)\( \beta_{2}) q^{15} +\)\(23\!\cdots\!16\)\( q^{16} +(\)\(49\!\cdots\!66\)\( + \)\(77\!\cdots\!78\)\( \beta_{1} + \)\(74\!\cdots\!30\)\( \beta_{2}) q^{17} +(\)\(21\!\cdots\!24\)\( - \)\(31\!\cdots\!48\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2}) q^{18} +(\)\(18\!\cdots\!00\)\( - \)\(10\!\cdots\!09\)\( \beta_{1} + \)\(28\!\cdots\!40\)\( \beta_{2}) q^{19} +(-\)\(14\!\cdots\!60\)\( + \)\(66\!\cdots\!88\)\( \beta_{1} - \)\(80\!\cdots\!28\)\( \beta_{2}) q^{20} +(\)\(12\!\cdots\!32\)\( + \)\(98\!\cdots\!68\)\( \beta_{1} + \)\(54\!\cdots\!40\)\( \beta_{2}) q^{21} +(-\)\(18\!\cdots\!76\)\( + \)\(29\!\cdots\!48\)\( \beta_{1} - \)\(17\!\cdots\!20\)\( \beta_{2}) q^{22} +(-\)\(26\!\cdots\!68\)\( - \)\(24\!\cdots\!86\)\( \beta_{1} - \)\(37\!\cdots\!60\)\( \beta_{2}) q^{23} +(-\)\(59\!\cdots\!64\)\( - \)\(10\!\cdots\!08\)\( \beta_{1}) q^{24} +(-\)\(16\!\cdots\!25\)\( + \)\(16\!\cdots\!60\)\( \beta_{1} - \)\(98\!\cdots\!60\)\( \beta_{2}) q^{25} +(-\)\(98\!\cdots\!16\)\( + \)\(36\!\cdots\!96\)\( \beta_{1} + \)\(79\!\cdots\!20\)\( \beta_{2}) q^{26} +(\)\(88\!\cdots\!20\)\( + \)\(93\!\cdots\!22\)\( \beta_{1} - \)\(94\!\cdots\!60\)\( \beta_{2}) q^{27} +(\)\(11\!\cdots\!84\)\( - \)\(13\!\cdots\!92\)\( \beta_{1} - \)\(59\!\cdots\!60\)\( \beta_{2}) q^{28} +(\)\(12\!\cdots\!10\)\( + \)\(43\!\cdots\!27\)\( \beta_{1} - \)\(38\!\cdots\!85\)\( \beta_{2}) q^{29} +(-\)\(11\!\cdots\!60\)\( + \)\(18\!\cdots\!48\)\( \beta_{1} + \)\(97\!\cdots\!12\)\( \beta_{2}) q^{30} +(\)\(91\!\cdots\!52\)\( + \)\(13\!\cdots\!48\)\( \beta_{1} + \)\(56\!\cdots\!40\)\( \beta_{2}) q^{31} +\)\(51\!\cdots\!32\)\( q^{32} +(-\)\(62\!\cdots\!96\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} - \)\(52\!\cdots\!30\)\( \beta_{2}) q^{33} +(\)\(10\!\cdots\!32\)\( + \)\(16\!\cdots\!56\)\( \beta_{1} + \)\(16\!\cdots\!60\)\( \beta_{2}) q^{34} +(\)\(48\!\cdots\!60\)\( + \)\(51\!\cdots\!12\)\( \beta_{1} - \)\(64\!\cdots\!72\)\( \beta_{2}) q^{35} +(\)\(46\!\cdots\!48\)\( - \)\(70\!\cdots\!96\)\( \beta_{1} + \)\(27\!\cdots\!60\)\( \beta_{2}) q^{36} +(-\)\(17\!\cdots\!34\)\( - \)\(17\!\cdots\!37\)\( \beta_{1} - \)\(52\!\cdots\!45\)\( \beta_{2}) q^{37} +(\)\(40\!\cdots\!00\)\( - \)\(23\!\cdots\!68\)\( \beta_{1} + \)\(62\!\cdots\!80\)\( \beta_{2}) q^{38} +(-\)\(80\!\cdots\!36\)\( - \)\(37\!\cdots\!06\)\( \beta_{1} - \)\(20\!\cdots\!00\)\( \beta_{2}) q^{39} +(-\)\(31\!\cdots\!20\)\( + \)\(14\!\cdots\!76\)\( \beta_{1} - \)\(17\!\cdots\!56\)\( \beta_{2}) q^{40} +(-\)\(54\!\cdots\!38\)\( - \)\(19\!\cdots\!28\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2}) q^{41} +(\)\(26\!\cdots\!64\)\( + \)\(21\!\cdots\!36\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2}) q^{42} +(-\)\(11\!\cdots\!68\)\( + \)\(84\!\cdots\!37\)\( \beta_{1} - \)\(38\!\cdots\!60\)\( \beta_{2}) q^{43} +(-\)\(40\!\cdots\!52\)\( + \)\(65\!\cdots\!96\)\( \beta_{1} - \)\(37\!\cdots\!40\)\( \beta_{2}) q^{44} +(-\)\(29\!\cdots\!30\)\( + \)\(26\!\cdots\!39\)\( \beta_{1} + \)\(41\!\cdots\!91\)\( \beta_{2}) q^{45} +(-\)\(59\!\cdots\!36\)\( - \)\(52\!\cdots\!72\)\( \beta_{1} - \)\(82\!\cdots\!20\)\( \beta_{2}) q^{46} +(-\)\(15\!\cdots\!04\)\( - \)\(36\!\cdots\!08\)\( \beta_{1} + \)\(34\!\cdots\!20\)\( \beta_{2}) q^{47} +(-\)\(13\!\cdots\!28\)\( - \)\(23\!\cdots\!16\)\( \beta_{1}) q^{48} +(-\)\(86\!\cdots\!27\)\( + \)\(29\!\cdots\!24\)\( \beta_{1} - \)\(87\!\cdots\!20\)\( \beta_{2}) q^{49} +(-\)\(36\!\cdots\!00\)\( + \)\(36\!\cdots\!20\)\( \beta_{1} - \)\(21\!\cdots\!20\)\( \beta_{2}) q^{50} +(-\)\(38\!\cdots\!28\)\( + \)\(81\!\cdots\!30\)\( \beta_{1} - \)\(45\!\cdots\!60\)\( \beta_{2}) q^{51} +(-\)\(21\!\cdots\!32\)\( + \)\(81\!\cdots\!92\)\( \beta_{1} + \)\(17\!\cdots\!40\)\( \beta_{2}) q^{52} +(\)\(65\!\cdots\!22\)\( + \)\(28\!\cdots\!43\)\( \beta_{1} - \)\(54\!\cdots\!05\)\( \beta_{2}) q^{53} +(\)\(19\!\cdots\!40\)\( + \)\(20\!\cdots\!44\)\( \beta_{1} - \)\(20\!\cdots\!20\)\( \beta_{2}) q^{54} +(\)\(70\!\cdots\!20\)\( - \)\(11\!\cdots\!86\)\( \beta_{1} + \)\(69\!\cdots\!16\)\( \beta_{2}) q^{55} +(\)\(25\!\cdots\!68\)\( - \)\(29\!\cdots\!84\)\( \beta_{1} - \)\(13\!\cdots\!20\)\( \beta_{2}) q^{56} +(\)\(51\!\cdots\!00\)\( - \)\(13\!\cdots\!38\)\( \beta_{1} - \)\(29\!\cdots\!10\)\( \beta_{2}) q^{57} +(\)\(27\!\cdots\!20\)\( + \)\(95\!\cdots\!04\)\( \beta_{1} - \)\(84\!\cdots\!20\)\( \beta_{2}) q^{58} +(\)\(58\!\cdots\!20\)\( + \)\(12\!\cdots\!09\)\( \beta_{1} + \)\(60\!\cdots\!80\)\( \beta_{2}) q^{59} +(-\)\(24\!\cdots\!20\)\( + \)\(41\!\cdots\!96\)\( \beta_{1} + \)\(21\!\cdots\!24\)\( \beta_{2}) q^{60} +(-\)\(14\!\cdots\!98\)\( - \)\(63\!\cdots\!49\)\( \beta_{1} - \)\(29\!\cdots\!45\)\( \beta_{2}) q^{61} +(\)\(20\!\cdots\!04\)\( + \)\(30\!\cdots\!96\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2}) q^{62} +(-\)\(15\!\cdots\!48\)\( - \)\(16\!\cdots\!10\)\( \beta_{1} + \)\(26\!\cdots\!40\)\( \beta_{2}) q^{63} +\)\(11\!\cdots\!64\)\( q^{64} +(-\)\(14\!\cdots\!80\)\( - \)\(22\!\cdots\!76\)\( \beta_{1} + \)\(11\!\cdots\!56\)\( \beta_{2}) q^{65} +(-\)\(13\!\cdots\!92\)\( + \)\(29\!\cdots\!12\)\( \beta_{1} - \)\(11\!\cdots\!60\)\( \beta_{2}) q^{66} +(\)\(47\!\cdots\!76\)\( - \)\(11\!\cdots\!49\)\( \beta_{1} - \)\(48\!\cdots\!80\)\( \beta_{2}) q^{67} +(\)\(24\!\cdots\!64\)\( + \)\(37\!\cdots\!12\)\( \beta_{1} + \)\(35\!\cdots\!20\)\( \beta_{2}) q^{68} +(\)\(26\!\cdots\!44\)\( + \)\(38\!\cdots\!16\)\( \beta_{1} + \)\(13\!\cdots\!20\)\( \beta_{2}) q^{69} +(\)\(10\!\cdots\!20\)\( + \)\(11\!\cdots\!24\)\( \beta_{1} - \)\(14\!\cdots\!44\)\( \beta_{2}) q^{70} +(\)\(60\!\cdots\!32\)\( - \)\(33\!\cdots\!74\)\( \beta_{1} + \)\(26\!\cdots\!40\)\( \beta_{2}) q^{71} +(\)\(10\!\cdots\!96\)\( - \)\(15\!\cdots\!92\)\( \beta_{1} + \)\(59\!\cdots\!20\)\( \beta_{2}) q^{72} +(-\)\(87\!\cdots\!38\)\( + \)\(41\!\cdots\!58\)\( \beta_{1} - \)\(18\!\cdots\!30\)\( \beta_{2}) q^{73} +(-\)\(38\!\cdots\!68\)\( - \)\(39\!\cdots\!24\)\( \beta_{1} - \)\(11\!\cdots\!40\)\( \beta_{2}) q^{74} +(-\)\(73\!\cdots\!00\)\( + \)\(20\!\cdots\!45\)\( \beta_{1} - \)\(94\!\cdots\!20\)\( \beta_{2}) q^{75} +(\)\(89\!\cdots\!00\)\( - \)\(51\!\cdots\!36\)\( \beta_{1} + \)\(13\!\cdots\!60\)\( \beta_{2}) q^{76} +(-\)\(11\!\cdots\!48\)\( + \)\(32\!\cdots\!08\)\( \beta_{1} - \)\(56\!\cdots\!20\)\( \beta_{2}) q^{77} +(-\)\(17\!\cdots\!72\)\( - \)\(82\!\cdots\!12\)\( \beta_{1} - \)\(45\!\cdots\!00\)\( \beta_{2}) q^{78} +(-\)\(40\!\cdots\!60\)\( + \)\(82\!\cdots\!96\)\( \beta_{1} + \)\(90\!\cdots\!80\)\( \beta_{2}) q^{79} +(-\)\(69\!\cdots\!40\)\( + \)\(32\!\cdots\!52\)\( \beta_{1} - \)\(38\!\cdots\!12\)\( \beta_{2}) q^{80} +(-\)\(89\!\cdots\!59\)\( + \)\(18\!\cdots\!82\)\( \beta_{1} - \)\(24\!\cdots\!30\)\( \beta_{2}) q^{81} +(-\)\(11\!\cdots\!76\)\( - \)\(42\!\cdots\!56\)\( \beta_{1} + \)\(24\!\cdots\!60\)\( \beta_{2}) q^{82} +(-\)\(28\!\cdots\!28\)\( - \)\(67\!\cdots\!85\)\( \beta_{1} + \)\(22\!\cdots\!20\)\( \beta_{2}) q^{83} +(\)\(58\!\cdots\!28\)\( + \)\(47\!\cdots\!72\)\( \beta_{1} + \)\(26\!\cdots\!60\)\( \beta_{2}) q^{84} +(\)\(33\!\cdots\!60\)\( - \)\(67\!\cdots\!98\)\( \beta_{1} - \)\(33\!\cdots\!62\)\( \beta_{2}) q^{85} +(-\)\(24\!\cdots\!36\)\( + \)\(18\!\cdots\!24\)\( \beta_{1} - \)\(83\!\cdots\!20\)\( \beta_{2}) q^{86} +(-\)\(22\!\cdots\!80\)\( + \)\(83\!\cdots\!54\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2}) q^{87} +(-\)\(88\!\cdots\!04\)\( + \)\(14\!\cdots\!92\)\( \beta_{1} - \)\(83\!\cdots\!80\)\( \beta_{2}) q^{88} +(-\)\(96\!\cdots\!90\)\( + \)\(20\!\cdots\!30\)\( \beta_{1} + \)\(59\!\cdots\!50\)\( \beta_{2}) q^{89} +(-\)\(65\!\cdots\!60\)\( + \)\(59\!\cdots\!28\)\( \beta_{1} + \)\(90\!\cdots\!32\)\( \beta_{2}) q^{90} +(-\)\(15\!\cdots\!68\)\( - \)\(99\!\cdots\!48\)\( \beta_{1} + \)\(18\!\cdots\!20\)\( \beta_{2}) q^{91} +(-\)\(12\!\cdots\!72\)\( - \)\(11\!\cdots\!44\)\( \beta_{1} - \)\(18\!\cdots\!40\)\( \beta_{2}) q^{92} +(-\)\(68\!\cdots\!16\)\( + \)\(63\!\cdots\!84\)\( \beta_{1} - \)\(95\!\cdots\!40\)\( \beta_{2}) q^{93} +(-\)\(33\!\cdots\!08\)\( - \)\(80\!\cdots\!16\)\( \beta_{1} + \)\(76\!\cdots\!40\)\( \beta_{2}) q^{94} +(-\)\(14\!\cdots\!00\)\( + \)\(37\!\cdots\!50\)\( \beta_{1} + \)\(82\!\cdots\!00\)\( \beta_{2}) q^{95} +(-\)\(28\!\cdots\!56\)\( - \)\(51\!\cdots\!32\)\( \beta_{1}) q^{96} +(-\)\(12\!\cdots\!94\)\( + \)\(29\!\cdots\!50\)\( \beta_{1} + \)\(91\!\cdots\!70\)\( \beta_{2}) q^{97} +(-\)\(18\!\cdots\!04\)\( + \)\(63\!\cdots\!48\)\( \beta_{1} - \)\(19\!\cdots\!40\)\( \beta_{2}) q^{98} +(-\)\(30\!\cdots\!56\)\( + \)\(51\!\cdots\!65\)\( \beta_{1} - \)\(28\!\cdots\!80\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 6597069766656q^{2} - 16865576978405999724q^{3} + \)\(14\!\cdots\!12\)\(q^{4} - \)\(89\!\cdots\!70\)\(q^{5} - \)\(37\!\cdots\!48\)\(q^{6} + \)\(72\!\cdots\!88\)\(q^{7} + \)\(31\!\cdots\!24\)\(q^{8} + \)\(29\!\cdots\!11\)\(q^{9} + O(q^{10}) \) \( 3q + 6597069766656q^{2} - 16865576978405999724q^{3} + \)\(14\!\cdots\!12\)\(q^{4} - \)\(89\!\cdots\!70\)\(q^{5} - \)\(37\!\cdots\!48\)\(q^{6} + \)\(72\!\cdots\!88\)\(q^{7} + \)\(31\!\cdots\!24\)\(q^{8} + \)\(29\!\cdots\!11\)\(q^{9} - \)\(19\!\cdots\!40\)\(q^{10} - \)\(24\!\cdots\!64\)\(q^{11} - \)\(81\!\cdots\!96\)\(q^{12} - \)\(13\!\cdots\!74\)\(q^{13} + \)\(16\!\cdots\!76\)\(q^{14} - \)\(15\!\cdots\!40\)\(q^{15} + \)\(70\!\cdots\!48\)\(q^{16} + \)\(14\!\cdots\!98\)\(q^{17} + \)\(63\!\cdots\!72\)\(q^{18} + \)\(55\!\cdots\!00\)\(q^{19} - \)\(43\!\cdots\!80\)\(q^{20} + \)\(36\!\cdots\!96\)\(q^{21} - \)\(54\!\cdots\!28\)\(q^{22} - \)\(80\!\cdots\!04\)\(q^{23} - \)\(17\!\cdots\!92\)\(q^{24} - \)\(50\!\cdots\!75\)\(q^{25} - \)\(29\!\cdots\!48\)\(q^{26} + \)\(26\!\cdots\!60\)\(q^{27} + \)\(35\!\cdots\!52\)\(q^{28} + \)\(38\!\cdots\!30\)\(q^{29} - \)\(33\!\cdots\!80\)\(q^{30} + \)\(27\!\cdots\!56\)\(q^{31} + \)\(15\!\cdots\!96\)\(q^{32} - \)\(18\!\cdots\!88\)\(q^{33} + \)\(32\!\cdots\!96\)\(q^{34} + \)\(14\!\cdots\!80\)\(q^{35} + \)\(14\!\cdots\!44\)\(q^{36} - \)\(52\!\cdots\!02\)\(q^{37} + \)\(12\!\cdots\!00\)\(q^{38} - \)\(24\!\cdots\!08\)\(q^{39} - \)\(94\!\cdots\!60\)\(q^{40} - \)\(16\!\cdots\!14\)\(q^{41} + \)\(80\!\cdots\!92\)\(q^{42} - \)\(33\!\cdots\!04\)\(q^{43} - \)\(12\!\cdots\!56\)\(q^{44} - \)\(89\!\cdots\!90\)\(q^{45} - \)\(17\!\cdots\!08\)\(q^{46} - \)\(46\!\cdots\!12\)\(q^{47} - \)\(39\!\cdots\!84\)\(q^{48} - \)\(25\!\cdots\!81\)\(q^{49} - \)\(11\!\cdots\!00\)\(q^{50} - \)\(11\!\cdots\!84\)\(q^{51} - \)\(64\!\cdots\!96\)\(q^{52} + \)\(19\!\cdots\!66\)\(q^{53} + \)\(58\!\cdots\!20\)\(q^{54} + \)\(21\!\cdots\!60\)\(q^{55} + \)\(77\!\cdots\!04\)\(q^{56} + \)\(15\!\cdots\!00\)\(q^{57} + \)\(83\!\cdots\!60\)\(q^{58} + \)\(17\!\cdots\!60\)\(q^{59} - \)\(74\!\cdots\!60\)\(q^{60} - \)\(43\!\cdots\!94\)\(q^{61} + \)\(60\!\cdots\!12\)\(q^{62} - \)\(45\!\cdots\!44\)\(q^{63} + \)\(33\!\cdots\!92\)\(q^{64} - \)\(42\!\cdots\!40\)\(q^{65} - \)\(41\!\cdots\!76\)\(q^{66} + \)\(14\!\cdots\!28\)\(q^{67} + \)\(72\!\cdots\!92\)\(q^{68} + \)\(80\!\cdots\!32\)\(q^{69} + \)\(31\!\cdots\!60\)\(q^{70} + \)\(18\!\cdots\!96\)\(q^{71} + \)\(30\!\cdots\!88\)\(q^{72} - \)\(26\!\cdots\!14\)\(q^{73} - \)\(11\!\cdots\!04\)\(q^{74} - \)\(21\!\cdots\!00\)\(q^{75} + \)\(26\!\cdots\!00\)\(q^{76} - \)\(34\!\cdots\!44\)\(q^{77} - \)\(52\!\cdots\!16\)\(q^{78} - \)\(12\!\cdots\!80\)\(q^{79} - \)\(20\!\cdots\!20\)\(q^{80} - \)\(26\!\cdots\!77\)\(q^{81} - \)\(35\!\cdots\!28\)\(q^{82} - \)\(86\!\cdots\!84\)\(q^{83} + \)\(17\!\cdots\!84\)\(q^{84} + \)\(99\!\cdots\!80\)\(q^{85} - \)\(73\!\cdots\!08\)\(q^{86} - \)\(66\!\cdots\!40\)\(q^{87} - \)\(26\!\cdots\!12\)\(q^{88} - \)\(29\!\cdots\!70\)\(q^{89} - \)\(19\!\cdots\!80\)\(q^{90} - \)\(47\!\cdots\!04\)\(q^{91} - \)\(38\!\cdots\!16\)\(q^{92} - \)\(20\!\cdots\!48\)\(q^{93} - \)\(10\!\cdots\!24\)\(q^{94} - \)\(43\!\cdots\!00\)\(q^{95} - \)\(86\!\cdots\!68\)\(q^{96} - \)\(37\!\cdots\!82\)\(q^{97} - \)\(56\!\cdots\!12\)\(q^{98} - \)\(90\!\cdots\!68\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 76392209863211857938006422774 x + 4214151671129618412000783695211690286445664\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 311040 \nu - 103680 \)
\(\beta_{2}\)\(=\)\((\)\( 163840 \nu^{2} + 13557284114987572480 \nu - 8344066442659091722136686534106880 \)\()/95350401\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 103680\)\()/311040\)
\(\nu^{2}\)\(=\)\((\)\(23170147443 \beta_{2} - 10591628214834041 \beta_{1} + 2027608145566158190339201513794600960\)\()/39813120\)

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
2.43002e14
5.76762e13
−3.00679e14
2.19902e12 −8.12053e19 4.83570e24 −9.96318e28 −1.78572e32 −5.66053e34 1.06338e37 2.60347e39 −2.19093e41
1.2 2.19902e12 −2.35615e19 4.83570e24 9.47163e28 −5.18122e31 1.10671e35 1.06338e37 −3.43570e39 2.08283e41
1.3 2.19902e12 8.79012e19 4.83570e24 −8.42086e28 1.93297e32 1.88466e34 1.06338e37 3.73579e39 −1.85177e41
\(n\): 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 \( T_{3}^{3} + \)\(16\!\cdots\!24\)\( T_{3}^{2} - \)\(72\!\cdots\!08\)\( T_{3} - \)\(16\!\cdots\!88\)\( \) acting on \(S_{84}^{\mathrm{new}}(\Gamma_0(2))\).