Properties

Label 2.84.a.a
Level 2
Weight 84
Character orbit 2.a
Self dual Yes
Analytic conductor 87.254
Analytic rank 0
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: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{7}\cdot 5^{3}\cdot 7\cdot 11\cdot 17 \)
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.

\(f(q)\) \(=\) \( q -2199023255552 q^{2} +(-33900915428623558692 + \beta_{1}) q^{3} +\)\(48\!\cdots\!04\)\( q^{4} +(-\)\(10\!\cdots\!50\)\( + 664677925 \beta_{1} + 4475 \beta_{2}) q^{5} +(\)\(74\!\cdots\!84\)\( - 2199023255552 \beta_{1}) q^{6} +(-\)\(18\!\cdots\!96\)\( + 1356511627483222 \beta_{1} - 1240294972 \beta_{2}) q^{7} -\)\(10\!\cdots\!08\)\( q^{8} +(-\)\(43\!\cdots\!63\)\( - \)\(10\!\cdots\!50\)\( \beta_{1} + 4985651180178 \beta_{2}) q^{9} +O(q^{10})\) \( q -2199023255552 q^{2} +(-33900915428623558692 + \beta_{1}) q^{3} +\)\(48\!\cdots\!04\)\( q^{4} +(-\)\(10\!\cdots\!50\)\( + 664677925 \beta_{1} + 4475 \beta_{2}) q^{5} +(\)\(74\!\cdots\!84\)\( - 2199023255552 \beta_{1}) q^{6} +(-\)\(18\!\cdots\!96\)\( + 1356511627483222 \beta_{1} - 1240294972 \beta_{2}) q^{7} -\)\(10\!\cdots\!08\)\( q^{8} +(-\)\(43\!\cdots\!63\)\( - \)\(10\!\cdots\!50\)\( \beta_{1} + 4985651180178 \beta_{2}) q^{9} +(\)\(22\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} - 9840629068595200 \beta_{2}) q^{10} +(-\)\(49\!\cdots\!88\)\( - \)\(25\!\cdots\!77\)\( \beta_{1} + 312292913603305928 \beta_{2}) q^{11} +(-\)\(16\!\cdots\!68\)\( + \)\(48\!\cdots\!04\)\( \beta_{1}) q^{12} +(-\)\(41\!\cdots\!42\)\( + \)\(30\!\cdots\!01\)\( \beta_{1} - \)\(37\!\cdots\!97\)\( \beta_{2}) q^{13} +(\)\(41\!\cdots\!92\)\( - \)\(29\!\cdots\!44\)\( \beta_{1} + \)\(27\!\cdots\!44\)\( \beta_{2}) q^{14} +(\)\(16\!\cdots\!00\)\( - \)\(40\!\cdots\!50\)\( \beta_{1} + \)\(64\!\cdots\!00\)\( \beta_{2}) q^{15} +\)\(23\!\cdots\!16\)\( q^{16} +(\)\(10\!\cdots\!34\)\( - \)\(20\!\cdots\!26\)\( \beta_{1} + \)\(45\!\cdots\!94\)\( \beta_{2}) q^{17} +(\)\(95\!\cdots\!76\)\( + \)\(22\!\cdots\!00\)\( \beta_{1} - \)\(10\!\cdots\!56\)\( \beta_{2}) q^{18} +(-\)\(38\!\cdots\!00\)\( + \)\(13\!\cdots\!65\)\( \beta_{1} + \)\(49\!\cdots\!28\)\( \beta_{2}) q^{19} +(-\)\(48\!\cdots\!00\)\( + \)\(32\!\cdots\!00\)\( \beta_{1} + \)\(21\!\cdots\!00\)\( \beta_{2}) q^{20} +(\)\(39\!\cdots\!32\)\( - \)\(11\!\cdots\!28\)\( \beta_{1} + \)\(58\!\cdots\!88\)\( \beta_{2}) q^{21} +(\)\(10\!\cdots\!76\)\( + \)\(56\!\cdots\!04\)\( \beta_{1} - \)\(68\!\cdots\!56\)\( \beta_{2}) q^{22} +(\)\(64\!\cdots\!68\)\( - \)\(20\!\cdots\!38\)\( \beta_{1} - \)\(97\!\cdots\!28\)\( \beta_{2}) q^{23} +(\)\(36\!\cdots\!36\)\( - \)\(10\!\cdots\!08\)\( \beta_{1}) q^{24} +(\)\(43\!\cdots\!75\)\( + \)\(18\!\cdots\!00\)\( \beta_{1} + \)\(17\!\cdots\!00\)\( \beta_{2}) q^{25} +(\)\(91\!\cdots\!84\)\( - \)\(67\!\cdots\!52\)\( \beta_{1} + \)\(81\!\cdots\!44\)\( \beta_{2}) q^{26} +(-\)\(96\!\cdots\!20\)\( + \)\(25\!\cdots\!54\)\( \beta_{1} - \)\(50\!\cdots\!28\)\( \beta_{2}) q^{27} +(-\)\(91\!\cdots\!84\)\( + \)\(65\!\cdots\!88\)\( \beta_{1} - \)\(59\!\cdots\!88\)\( \beta_{2}) q^{28} +(-\)\(44\!\cdots\!90\)\( + \)\(16\!\cdots\!81\)\( \beta_{1} - \)\(10\!\cdots\!37\)\( \beta_{2}) q^{29} +(-\)\(35\!\cdots\!00\)\( + \)\(88\!\cdots\!00\)\( \beta_{1} - \)\(14\!\cdots\!00\)\( \beta_{2}) q^{30} +(-\)\(50\!\cdots\!48\)\( + \)\(18\!\cdots\!92\)\( \beta_{1} + \)\(22\!\cdots\!68\)\( \beta_{2}) q^{31} -\)\(51\!\cdots\!32\)\( q^{32} +(-\)\(44\!\cdots\!04\)\( + \)\(12\!\cdots\!22\)\( \beta_{1} - \)\(10\!\cdots\!34\)\( \beta_{2}) q^{33} +(-\)\(22\!\cdots\!68\)\( + \)\(45\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2}) q^{34} +(-\)\(15\!\cdots\!00\)\( - \)\(91\!\cdots\!00\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2}) q^{35} +(-\)\(20\!\cdots\!52\)\( - \)\(49\!\cdots\!00\)\( \beta_{1} + \)\(24\!\cdots\!12\)\( \beta_{2}) q^{36} +(-\)\(48\!\cdots\!66\)\( + \)\(14\!\cdots\!09\)\( \beta_{1} + \)\(41\!\cdots\!59\)\( \beta_{2}) q^{37} +(\)\(84\!\cdots\!00\)\( - \)\(29\!\cdots\!80\)\( \beta_{1} - \)\(10\!\cdots\!56\)\( \beta_{2}) q^{38} +(\)\(75\!\cdots\!64\)\( - \)\(21\!\cdots\!06\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2}) q^{39} +(\)\(10\!\cdots\!00\)\( - \)\(70\!\cdots\!00\)\( \beta_{1} - \)\(47\!\cdots\!00\)\( \beta_{2}) q^{40} +(-\)\(49\!\cdots\!38\)\( + \)\(75\!\cdots\!00\)\( \beta_{1} + \)\(17\!\cdots\!16\)\( \beta_{2}) q^{41} +(-\)\(86\!\cdots\!64\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} - \)\(12\!\cdots\!76\)\( \beta_{2}) q^{42} +(-\)\(61\!\cdots\!32\)\( + \)\(53\!\cdots\!79\)\( \beta_{1} + \)\(24\!\cdots\!52\)\( \beta_{2}) q^{43} +(-\)\(24\!\cdots\!52\)\( - \)\(12\!\cdots\!08\)\( \beta_{1} + \)\(15\!\cdots\!12\)\( \beta_{2}) q^{44} +(-\)\(14\!\cdots\!50\)\( + \)\(17\!\cdots\!25\)\( \beta_{1} - \)\(18\!\cdots\!25\)\( \beta_{2}) q^{45} +(-\)\(14\!\cdots\!36\)\( + \)\(45\!\cdots\!76\)\( \beta_{1} + \)\(21\!\cdots\!56\)\( \beta_{2}) q^{46} +(-\)\(11\!\cdots\!96\)\( - \)\(37\!\cdots\!24\)\( \beta_{1} + \)\(40\!\cdots\!96\)\( \beta_{2}) q^{47} +(-\)\(79\!\cdots\!72\)\( + \)\(23\!\cdots\!16\)\( \beta_{1}) q^{48} +(-\)\(80\!\cdots\!27\)\( - \)\(10\!\cdots\!28\)\( \beta_{1} - \)\(62\!\cdots\!44\)\( \beta_{2}) q^{49} +(-\)\(94\!\cdots\!00\)\( - \)\(41\!\cdots\!00\)\( \beta_{1} - \)\(37\!\cdots\!00\)\( \beta_{2}) q^{50} +(-\)\(53\!\cdots\!28\)\( + \)\(15\!\cdots\!54\)\( \beta_{1} - \)\(70\!\cdots\!72\)\( \beta_{2}) q^{51} +(-\)\(20\!\cdots\!68\)\( + \)\(14\!\cdots\!04\)\( \beta_{1} - \)\(17\!\cdots\!88\)\( \beta_{2}) q^{52} +(-\)\(12\!\cdots\!22\)\( - \)\(98\!\cdots\!55\)\( \beta_{1} + \)\(13\!\cdots\!11\)\( \beta_{2}) q^{53} +(\)\(21\!\cdots\!40\)\( - \)\(57\!\cdots\!08\)\( \beta_{1} + \)\(11\!\cdots\!56\)\( \beta_{2}) q^{54} +(\)\(54\!\cdots\!00\)\( + \)\(15\!\cdots\!50\)\( \beta_{1} - \)\(57\!\cdots\!00\)\( \beta_{2}) q^{55} +(\)\(20\!\cdots\!68\)\( - \)\(14\!\cdots\!76\)\( \beta_{1} + \)\(13\!\cdots\!76\)\( \beta_{2}) q^{56} +(\)\(45\!\cdots\!00\)\( - \)\(12\!\cdots\!26\)\( \beta_{1} + \)\(10\!\cdots\!42\)\( \beta_{2}) q^{57} +(\)\(98\!\cdots\!80\)\( - \)\(35\!\cdots\!12\)\( \beta_{1} + \)\(23\!\cdots\!24\)\( \beta_{2}) q^{58} +(\)\(10\!\cdots\!20\)\( + \)\(41\!\cdots\!67\)\( \beta_{1} - \)\(51\!\cdots\!24\)\( \beta_{2}) q^{59} +(\)\(79\!\cdots\!00\)\( - \)\(19\!\cdots\!00\)\( \beta_{1} + \)\(31\!\cdots\!00\)\( \beta_{2}) q^{60} +(-\)\(10\!\cdots\!98\)\( + \)\(19\!\cdots\!09\)\( \beta_{1} - \)\(14\!\cdots\!49\)\( \beta_{2}) q^{61} +(\)\(11\!\cdots\!96\)\( - \)\(39\!\cdots\!84\)\( \beta_{1} - \)\(48\!\cdots\!36\)\( \beta_{2}) q^{62} +(-\)\(33\!\cdots\!52\)\( + \)\(62\!\cdots\!86\)\( \beta_{1} + \)\(43\!\cdots\!72\)\( \beta_{2}) q^{63} +\)\(11\!\cdots\!64\)\( q^{64} +(-\)\(63\!\cdots\!00\)\( - \)\(22\!\cdots\!00\)\( \beta_{1} + \)\(40\!\cdots\!00\)\( \beta_{2}) q^{65} +(\)\(97\!\cdots\!08\)\( - \)\(28\!\cdots\!44\)\( \beta_{1} + \)\(23\!\cdots\!68\)\( \beta_{2}) q^{66} +(-\)\(32\!\cdots\!76\)\( + \)\(69\!\cdots\!57\)\( \beta_{1} - \)\(24\!\cdots\!24\)\( \beta_{2}) q^{67} +(\)\(49\!\cdots\!36\)\( - \)\(99\!\cdots\!04\)\( \beta_{1} + \)\(22\!\cdots\!76\)\( \beta_{2}) q^{68} +(-\)\(72\!\cdots\!56\)\( + \)\(19\!\cdots\!28\)\( \beta_{1} - \)\(17\!\cdots\!36\)\( \beta_{2}) q^{69} +(\)\(34\!\cdots\!00\)\( + \)\(20\!\cdots\!00\)\( \beta_{1} - \)\(24\!\cdots\!00\)\( \beta_{2}) q^{70} +(\)\(27\!\cdots\!32\)\( - \)\(15\!\cdots\!90\)\( \beta_{1} + \)\(15\!\cdots\!48\)\( \beta_{2}) q^{71} +(\)\(45\!\cdots\!04\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} - \)\(53\!\cdots\!24\)\( \beta_{2}) q^{72} +(\)\(18\!\cdots\!38\)\( - \)\(10\!\cdots\!70\)\( \beta_{1} - \)\(15\!\cdots\!14\)\( \beta_{2}) q^{73} +(\)\(10\!\cdots\!32\)\( - \)\(32\!\cdots\!68\)\( \beta_{1} - \)\(91\!\cdots\!68\)\( \beta_{2}) q^{74} +(\)\(30\!\cdots\!00\)\( - \)\(82\!\cdots\!25\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2}) q^{75} +(-\)\(18\!\cdots\!00\)\( + \)\(65\!\cdots\!60\)\( \beta_{1} + \)\(24\!\cdots\!12\)\( \beta_{2}) q^{76} +(-\)\(10\!\cdots\!52\)\( + \)\(73\!\cdots\!44\)\( \beta_{1} + \)\(24\!\cdots\!44\)\( \beta_{2}) q^{77} +(-\)\(16\!\cdots\!28\)\( + \)\(48\!\cdots\!12\)\( \beta_{1} - \)\(27\!\cdots\!00\)\( \beta_{2}) q^{78} +(-\)\(13\!\cdots\!60\)\( - \)\(50\!\cdots\!76\)\( \beta_{1} - \)\(16\!\cdots\!84\)\( \beta_{2}) q^{79} +(-\)\(23\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2}) q^{80} +(\)\(11\!\cdots\!41\)\( + \)\(13\!\cdots\!54\)\( \beta_{1} - \)\(72\!\cdots\!66\)\( \beta_{2}) q^{81} +(\)\(10\!\cdots\!76\)\( - \)\(16\!\cdots\!00\)\( \beta_{1} - \)\(39\!\cdots\!32\)\( \beta_{2}) q^{82} +(\)\(17\!\cdots\!28\)\( - \)\(42\!\cdots\!67\)\( \beta_{1} + \)\(14\!\cdots\!56\)\( \beta_{2}) q^{83} +(\)\(18\!\cdots\!28\)\( - \)\(54\!\cdots\!12\)\( \beta_{1} + \)\(28\!\cdots\!52\)\( \beta_{2}) q^{84} +(\)\(10\!\cdots\!00\)\( + \)\(24\!\cdots\!50\)\( \beta_{1} - \)\(20\!\cdots\!50\)\( \beta_{2}) q^{85} +(\)\(13\!\cdots\!64\)\( - \)\(11\!\cdots\!08\)\( \beta_{1} - \)\(53\!\cdots\!04\)\( \beta_{2}) q^{86} +(\)\(19\!\cdots\!80\)\( - \)\(57\!\cdots\!14\)\( \beta_{1} + \)\(53\!\cdots\!80\)\( \beta_{2}) q^{87} +(\)\(52\!\cdots\!04\)\( + \)\(27\!\cdots\!16\)\( \beta_{1} - \)\(33\!\cdots\!24\)\( \beta_{2}) q^{88} +(\)\(24\!\cdots\!10\)\( + \)\(23\!\cdots\!30\)\( \beta_{1} + \)\(10\!\cdots\!50\)\( \beta_{2}) q^{89} +(\)\(32\!\cdots\!00\)\( - \)\(38\!\cdots\!00\)\( \beta_{1} + \)\(39\!\cdots\!00\)\( \beta_{2}) q^{90} +(\)\(13\!\cdots\!32\)\( - \)\(17\!\cdots\!56\)\( \beta_{1} - \)\(21\!\cdots\!76\)\( \beta_{2}) q^{91} +(\)\(31\!\cdots\!72\)\( - \)\(10\!\cdots\!52\)\( \beta_{1} - \)\(46\!\cdots\!12\)\( \beta_{2}) q^{92} +(\)\(45\!\cdots\!16\)\( - \)\(12\!\cdots\!20\)\( \beta_{1} + \)\(10\!\cdots\!08\)\( \beta_{2}) q^{93} +(\)\(25\!\cdots\!92\)\( + \)\(83\!\cdots\!48\)\( \beta_{1} - \)\(90\!\cdots\!92\)\( \beta_{2}) q^{94} +(\)\(17\!\cdots\!00\)\( + \)\(17\!\cdots\!50\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2}) q^{95} +(\)\(17\!\cdots\!44\)\( - \)\(51\!\cdots\!32\)\( \beta_{1}) q^{96} +(\)\(39\!\cdots\!94\)\( - \)\(40\!\cdots\!82\)\( \beta_{1} + \)\(43\!\cdots\!46\)\( \beta_{2}) q^{97} +(\)\(17\!\cdots\!04\)\( + \)\(23\!\cdots\!56\)\( \beta_{1} + \)\(13\!\cdots\!88\)\( \beta_{2}) q^{98} +(\)\(66\!\cdots\!44\)\( - \)\(31\!\cdots\!33\)\( \beta_{1} - \)\(11\!\cdots\!56\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 6597069766656q^{2} - \)\(10\!\cdots\!76\)\(q^{3} + \)\(14\!\cdots\!12\)\(q^{4} - \)\(30\!\cdots\!50\)\(q^{5} + \)\(22\!\cdots\!52\)\(q^{6} - \)\(56\!\cdots\!88\)\(q^{7} - \)\(31\!\cdots\!24\)\(q^{8} - \)\(12\!\cdots\!89\)\(q^{9} + O(q^{10}) \) \( 3q - 6597069766656q^{2} - \)\(10\!\cdots\!76\)\(q^{3} + \)\(14\!\cdots\!12\)\(q^{4} - \)\(30\!\cdots\!50\)\(q^{5} + \)\(22\!\cdots\!52\)\(q^{6} - \)\(56\!\cdots\!88\)\(q^{7} - \)\(31\!\cdots\!24\)\(q^{8} - \)\(12\!\cdots\!89\)\(q^{9} + \)\(66\!\cdots\!00\)\(q^{10} - \)\(14\!\cdots\!64\)\(q^{11} - \)\(49\!\cdots\!04\)\(q^{12} - \)\(12\!\cdots\!26\)\(q^{13} + \)\(12\!\cdots\!76\)\(q^{14} + \)\(49\!\cdots\!00\)\(q^{15} + \)\(70\!\cdots\!48\)\(q^{16} + \)\(30\!\cdots\!02\)\(q^{17} + \)\(28\!\cdots\!28\)\(q^{18} - \)\(11\!\cdots\!00\)\(q^{19} - \)\(14\!\cdots\!00\)\(q^{20} + \)\(11\!\cdots\!96\)\(q^{21} + \)\(32\!\cdots\!28\)\(q^{22} + \)\(19\!\cdots\!04\)\(q^{23} + \)\(10\!\cdots\!08\)\(q^{24} + \)\(12\!\cdots\!25\)\(q^{25} + \)\(27\!\cdots\!52\)\(q^{26} - \)\(29\!\cdots\!60\)\(q^{27} - \)\(27\!\cdots\!52\)\(q^{28} - \)\(13\!\cdots\!70\)\(q^{29} - \)\(10\!\cdots\!00\)\(q^{30} - \)\(15\!\cdots\!44\)\(q^{31} - \)\(15\!\cdots\!96\)\(q^{32} - \)\(13\!\cdots\!12\)\(q^{33} - \)\(67\!\cdots\!04\)\(q^{34} - \)\(47\!\cdots\!00\)\(q^{35} - \)\(62\!\cdots\!56\)\(q^{36} - \)\(14\!\cdots\!98\)\(q^{37} + \)\(25\!\cdots\!00\)\(q^{38} + \)\(22\!\cdots\!92\)\(q^{39} + \)\(32\!\cdots\!00\)\(q^{40} - \)\(14\!\cdots\!14\)\(q^{41} - \)\(25\!\cdots\!92\)\(q^{42} - \)\(18\!\cdots\!96\)\(q^{43} - \)\(72\!\cdots\!56\)\(q^{44} - \)\(44\!\cdots\!50\)\(q^{45} - \)\(42\!\cdots\!08\)\(q^{46} - \)\(34\!\cdots\!88\)\(q^{47} - \)\(23\!\cdots\!16\)\(q^{48} - \)\(24\!\cdots\!81\)\(q^{49} - \)\(28\!\cdots\!00\)\(q^{50} - \)\(15\!\cdots\!84\)\(q^{51} - \)\(60\!\cdots\!04\)\(q^{52} - \)\(38\!\cdots\!66\)\(q^{53} + \)\(63\!\cdots\!20\)\(q^{54} + \)\(16\!\cdots\!00\)\(q^{55} + \)\(60\!\cdots\!04\)\(q^{56} + \)\(13\!\cdots\!00\)\(q^{57} + \)\(29\!\cdots\!40\)\(q^{58} + \)\(31\!\cdots\!60\)\(q^{59} + \)\(23\!\cdots\!00\)\(q^{60} - \)\(30\!\cdots\!94\)\(q^{61} + \)\(33\!\cdots\!88\)\(q^{62} - \)\(99\!\cdots\!56\)\(q^{63} + \)\(33\!\cdots\!92\)\(q^{64} - \)\(19\!\cdots\!00\)\(q^{65} + \)\(29\!\cdots\!24\)\(q^{66} - \)\(98\!\cdots\!28\)\(q^{67} + \)\(14\!\cdots\!08\)\(q^{68} - \)\(21\!\cdots\!68\)\(q^{69} + \)\(10\!\cdots\!00\)\(q^{70} + \)\(81\!\cdots\!96\)\(q^{71} + \)\(13\!\cdots\!12\)\(q^{72} + \)\(54\!\cdots\!14\)\(q^{73} + \)\(31\!\cdots\!96\)\(q^{74} + \)\(91\!\cdots\!00\)\(q^{75} - \)\(55\!\cdots\!00\)\(q^{76} - \)\(30\!\cdots\!56\)\(q^{77} - \)\(49\!\cdots\!84\)\(q^{78} - \)\(40\!\cdots\!80\)\(q^{79} - \)\(70\!\cdots\!00\)\(q^{80} + \)\(33\!\cdots\!23\)\(q^{81} + \)\(32\!\cdots\!28\)\(q^{82} + \)\(52\!\cdots\!84\)\(q^{83} + \)\(56\!\cdots\!84\)\(q^{84} + \)\(31\!\cdots\!00\)\(q^{85} + \)\(40\!\cdots\!92\)\(q^{86} + \)\(57\!\cdots\!40\)\(q^{87} + \)\(15\!\cdots\!12\)\(q^{88} + \)\(73\!\cdots\!30\)\(q^{89} + \)\(97\!\cdots\!00\)\(q^{90} + \)\(39\!\cdots\!96\)\(q^{91} + \)\(94\!\cdots\!16\)\(q^{92} + \)\(13\!\cdots\!48\)\(q^{93} + \)\(75\!\cdots\!76\)\(q^{94} + \)\(51\!\cdots\!00\)\(q^{95} + \)\(52\!\cdots\!32\)\(q^{96} + \)\(11\!\cdots\!82\)\(q^{97} + \)\(53\!\cdots\!12\)\(q^{98} + \)\(19\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 287609867501924274375802127400 x - 41230865304567060522794640394926417995512500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -18 \nu^{2} + 8140017811031118 \nu + 3451318410023088579170355185100 \)\()/ 52252679285 \)
\(\beta_{2}\)\(=\)\((\)\( 10166454 \nu^{2} - 58893267023738651754 \nu - 1949314991936272011652225019633331300 \)\()/ 52252679285 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 564803 \beta_{1} + 28952985600\)\()/ 86858956800 \)
\(\nu^{2}\)\(=\)\((\)\(452223211723951 \beta_{2} + 3271848167985480653 \beta_{1} + 16654328704402242976586095966152083865600\)\()/ 86858956800 \)

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
−4.40454e14
−1.56747e14
5.97202e14
−2.19902e12 −1.03294e20 4.83570e24 −4.29409e28 2.27146e32 −1.14250e35 −1.06338e37 6.67886e39 9.44280e40
1.2 −2.19902e12 −7.32556e17 4.83570e24 −1.23719e29 1.61091e30 6.61589e34 −1.06338e37 −3.99030e39 2.72062e41
1.3 −2.19902e12 2.32406e18 4.83570e24 1.63642e29 −5.11067e30 −8.77702e33 −1.06338e37 −3.98544e39 −3.59853e41
\(n\): 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 \( T_{3}^{3} + \)\(10\!\cdots\!76\)\( T_{3}^{2} - \)\(16\!\cdots\!08\)\( T_{3} - \)\(17\!\cdots\!12\)\( \) acting on \(S_{84}^{\mathrm{new}}(\Gamma_0(2))\).