Properties

Label 2.80.a.a
Level 2
Weight 80
Character orbit 2.a
Self dual yes
Analytic conductor 79.047
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 80 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.0474097443\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{9}\cdot 5^{4}\cdot 7\cdot 11 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q +549755813888 q^{2} +(-1528323113916241812 - \beta_{1}) q^{3} +\)\(30\!\cdots\!44\)\( q^{4} +(-\)\(13\!\cdots\!50\)\( - 44183620 \beta_{1} + 235 \beta_{2}) q^{5} +(-\)\(84\!\cdots\!56\)\( - 549755813888 \beta_{1}) q^{6} +(\)\(49\!\cdots\!04\)\( + 308388169933606 \beta_{1} - 64294732 \beta_{2}) q^{7} +\)\(16\!\cdots\!72\)\( q^{8} +(-\)\(22\!\cdots\!23\)\( - 289238058333167832 \beta_{1} - 1128071453022 \beta_{2}) q^{9} +O(q^{10})\) \( q +549755813888 q^{2} +(-1528323113916241812 - \beta_{1}) q^{3} +\)\(30\!\cdots\!44\)\( q^{4} +(-\)\(13\!\cdots\!50\)\( - 44183620 \beta_{1} + 235 \beta_{2}) q^{5} +(-\)\(84\!\cdots\!56\)\( - 549755813888 \beta_{1}) q^{6} +(\)\(49\!\cdots\!04\)\( + 308388169933606 \beta_{1} - 64294732 \beta_{2}) q^{7} +\)\(16\!\cdots\!72\)\( q^{8} +(-\)\(22\!\cdots\!23\)\( - 289238058333167832 \beta_{1} - 1128071453022 \beta_{2}) q^{9} +(-\)\(74\!\cdots\!00\)\( - 24290201973618114560 \beta_{1} + 129192616263680 \beta_{2}) q^{10} +(-\)\(41\!\cdots\!28\)\( - \)\(59\!\cdots\!35\)\( \beta_{1} + 53330002738408 \beta_{2}) q^{11} +(-\)\(46\!\cdots\!28\)\( - \)\(30\!\cdots\!44\)\( \beta_{1}) q^{12} +(\)\(14\!\cdots\!18\)\( - \)\(21\!\cdots\!68\)\( \beta_{1} + 3386116444972880203 \beta_{2}) q^{13} +(\)\(27\!\cdots\!52\)\( + \)\(16\!\cdots\!28\)\( \beta_{1} - 35346402719370838016 \beta_{2}) q^{14} +(\)\(40\!\cdots\!00\)\( + \)\(35\!\cdots\!90\)\( \beta_{1} - \)\(11\!\cdots\!20\)\( \beta_{2}) q^{15} +\)\(91\!\cdots\!36\)\( q^{16} +(\)\(31\!\cdots\!94\)\( - \)\(41\!\cdots\!00\)\( \beta_{1} - \)\(22\!\cdots\!26\)\( \beta_{2}) q^{17} +(-\)\(12\!\cdots\!24\)\( - \)\(15\!\cdots\!16\)\( \beta_{1} - \)\(62\!\cdots\!36\)\( \beta_{2}) q^{18} +(\)\(67\!\cdots\!00\)\( + \)\(27\!\cdots\!63\)\( \beta_{1} - \)\(93\!\cdots\!12\)\( \beta_{2}) q^{19} +(-\)\(40\!\cdots\!00\)\( - \)\(13\!\cdots\!80\)\( \beta_{1} + \)\(71\!\cdots\!40\)\( \beta_{2}) q^{20} +(-\)\(14\!\cdots\!48\)\( - \)\(57\!\cdots\!24\)\( \beta_{1} + \)\(66\!\cdots\!08\)\( \beta_{2}) q^{21} +(-\)\(22\!\cdots\!64\)\( - \)\(32\!\cdots\!80\)\( \beta_{1} + \)\(29\!\cdots\!04\)\( \beta_{2}) q^{22} +(\)\(27\!\cdots\!08\)\( + \)\(59\!\cdots\!50\)\( \beta_{1} - \)\(21\!\cdots\!08\)\( \beta_{2}) q^{23} +(-\)\(25\!\cdots\!64\)\( - \)\(16\!\cdots\!72\)\( \beta_{1}) q^{24} +(\)\(70\!\cdots\!75\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} - \)\(16\!\cdots\!00\)\( \beta_{2}) q^{25} +(\)\(78\!\cdots\!84\)\( - \)\(12\!\cdots\!84\)\( \beta_{1} + \)\(18\!\cdots\!64\)\( \beta_{2}) q^{26} +(\)\(91\!\cdots\!80\)\( + \)\(39\!\cdots\!18\)\( \beta_{1} + \)\(51\!\cdots\!92\)\( \beta_{2}) q^{27} +(\)\(14\!\cdots\!76\)\( + \)\(93\!\cdots\!64\)\( \beta_{1} - \)\(19\!\cdots\!08\)\( \beta_{2}) q^{28} +(\)\(49\!\cdots\!10\)\( + \)\(83\!\cdots\!92\)\( \beta_{1} - \)\(70\!\cdots\!17\)\( \beta_{2}) q^{29} +(\)\(22\!\cdots\!00\)\( + \)\(19\!\cdots\!20\)\( \beta_{1} - \)\(65\!\cdots\!60\)\( \beta_{2}) q^{30} +(-\)\(31\!\cdots\!68\)\( - \)\(42\!\cdots\!84\)\( \beta_{1} + \)\(42\!\cdots\!28\)\( \beta_{2}) q^{31} +\)\(50\!\cdots\!68\)\( q^{32} +(\)\(33\!\cdots\!36\)\( + \)\(31\!\cdots\!84\)\( \beta_{1} - \)\(70\!\cdots\!14\)\( \beta_{2}) q^{33} +(\)\(17\!\cdots\!72\)\( - \)\(23\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2}) q^{34} +(-\)\(72\!\cdots\!00\)\( - \)\(15\!\cdots\!80\)\( \beta_{1} + \)\(32\!\cdots\!40\)\( \beta_{2}) q^{35} +(-\)\(67\!\cdots\!12\)\( - \)\(87\!\cdots\!08\)\( \beta_{1} - \)\(34\!\cdots\!68\)\( \beta_{2}) q^{36} +(-\)\(48\!\cdots\!86\)\( + \)\(22\!\cdots\!80\)\( \beta_{1} + \)\(30\!\cdots\!59\)\( \beta_{2}) q^{37} +(\)\(37\!\cdots\!00\)\( + \)\(15\!\cdots\!44\)\( \beta_{1} - \)\(51\!\cdots\!56\)\( \beta_{2}) q^{38} +(\)\(76\!\cdots\!84\)\( + \)\(15\!\cdots\!26\)\( \beta_{1} - \)\(18\!\cdots\!00\)\( \beta_{2}) q^{39} +(-\)\(22\!\cdots\!00\)\( - \)\(73\!\cdots\!40\)\( \beta_{1} + \)\(39\!\cdots\!20\)\( \beta_{2}) q^{40} +(-\)\(29\!\cdots\!98\)\( + \)\(73\!\cdots\!96\)\( \beta_{1} - \)\(32\!\cdots\!84\)\( \beta_{2}) q^{41} +(-\)\(79\!\cdots\!24\)\( - \)\(31\!\cdots\!12\)\( \beta_{1} + \)\(36\!\cdots\!04\)\( \beta_{2}) q^{42} +(-\)\(31\!\cdots\!32\)\( - \)\(51\!\cdots\!67\)\( \beta_{1} - \)\(63\!\cdots\!88\)\( \beta_{2}) q^{43} +(-\)\(12\!\cdots\!32\)\( - \)\(18\!\cdots\!40\)\( \beta_{1} + \)\(16\!\cdots\!52\)\( \beta_{2}) q^{44} +(-\)\(10\!\cdots\!50\)\( - \)\(71\!\cdots\!40\)\( \beta_{1} - \)\(16\!\cdots\!05\)\( \beta_{2}) q^{45} +(\)\(15\!\cdots\!04\)\( + \)\(32\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!04\)\( \beta_{2}) q^{46} +(-\)\(39\!\cdots\!56\)\( + \)\(33\!\cdots\!40\)\( \beta_{1} + \)\(67\!\cdots\!36\)\( \beta_{2}) q^{47} +(-\)\(13\!\cdots\!32\)\( - \)\(91\!\cdots\!36\)\( \beta_{1}) q^{48} +(\)\(32\!\cdots\!73\)\( + \)\(50\!\cdots\!64\)\( \beta_{1} - \)\(26\!\cdots\!64\)\( \beta_{2}) q^{49} +(\)\(38\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} - \)\(88\!\cdots\!00\)\( \beta_{2}) q^{50} +(\)\(18\!\cdots\!72\)\( - \)\(33\!\cdots\!06\)\( \beta_{1} + \)\(63\!\cdots\!68\)\( \beta_{2}) q^{51} +(\)\(43\!\cdots\!92\)\( - \)\(66\!\cdots\!92\)\( \beta_{1} + \)\(10\!\cdots\!32\)\( \beta_{2}) q^{52} +(\)\(41\!\cdots\!58\)\( - \)\(18\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!49\)\( \beta_{2}) q^{53} +(\)\(50\!\cdots\!40\)\( + \)\(21\!\cdots\!84\)\( \beta_{1} + \)\(28\!\cdots\!96\)\( \beta_{2}) q^{54} +(\)\(72\!\cdots\!00\)\( + \)\(23\!\cdots\!10\)\( \beta_{1} - \)\(14\!\cdots\!80\)\( \beta_{2}) q^{55} +(\)\(82\!\cdots\!88\)\( + \)\(51\!\cdots\!32\)\( \beta_{1} - \)\(10\!\cdots\!04\)\( \beta_{2}) q^{56} +(-\)\(12\!\cdots\!00\)\( - \)\(48\!\cdots\!72\)\( \beta_{1} + \)\(76\!\cdots\!02\)\( \beta_{2}) q^{57} +(\)\(27\!\cdots\!80\)\( + \)\(45\!\cdots\!96\)\( \beta_{1} - \)\(38\!\cdots\!96\)\( \beta_{2}) q^{58} +(-\)\(39\!\cdots\!80\)\( - \)\(10\!\cdots\!31\)\( \beta_{1} + \)\(16\!\cdots\!76\)\( \beta_{2}) q^{59} +(\)\(12\!\cdots\!00\)\( + \)\(10\!\cdots\!60\)\( \beta_{1} - \)\(36\!\cdots\!80\)\( \beta_{2}) q^{60} +(-\)\(21\!\cdots\!38\)\( - \)\(44\!\cdots\!48\)\( \beta_{1} - \)\(98\!\cdots\!89\)\( \beta_{2}) q^{61} +(-\)\(17\!\cdots\!84\)\( - \)\(23\!\cdots\!92\)\( \beta_{1} + \)\(23\!\cdots\!64\)\( \beta_{2}) q^{62} +(\)\(23\!\cdots\!08\)\( + \)\(48\!\cdots\!86\)\( \beta_{1} - \)\(69\!\cdots\!28\)\( \beta_{2}) q^{63} +\)\(27\!\cdots\!84\)\( q^{64} +(\)\(29\!\cdots\!00\)\( + \)\(31\!\cdots\!40\)\( \beta_{1} + \)\(43\!\cdots\!80\)\( \beta_{2}) q^{65} +(\)\(18\!\cdots\!68\)\( + \)\(17\!\cdots\!92\)\( \beta_{1} - \)\(38\!\cdots\!32\)\( \beta_{2}) q^{66} +(\)\(35\!\cdots\!84\)\( + \)\(18\!\cdots\!59\)\( \beta_{1} - \)\(17\!\cdots\!04\)\( \beta_{2}) q^{67} +(\)\(95\!\cdots\!36\)\( - \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(68\!\cdots\!44\)\( \beta_{2}) q^{68} +(-\)\(27\!\cdots\!96\)\( - \)\(13\!\cdots\!04\)\( \beta_{1} + \)\(17\!\cdots\!44\)\( \beta_{2}) q^{69} +(-\)\(39\!\cdots\!00\)\( - \)\(86\!\cdots\!40\)\( \beta_{1} + \)\(17\!\cdots\!20\)\( \beta_{2}) q^{70} +(-\)\(39\!\cdots\!48\)\( + \)\(10\!\cdots\!98\)\( \beta_{1} - \)\(25\!\cdots\!32\)\( \beta_{2}) q^{71} +(-\)\(36\!\cdots\!56\)\( - \)\(48\!\cdots\!04\)\( \beta_{1} - \)\(18\!\cdots\!84\)\( \beta_{2}) q^{72} +(-\)\(70\!\cdots\!22\)\( + \)\(33\!\cdots\!96\)\( \beta_{1} - \)\(13\!\cdots\!54\)\( \beta_{2}) q^{73} +(-\)\(26\!\cdots\!68\)\( + \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(16\!\cdots\!92\)\( \beta_{2}) q^{74} +(-\)\(97\!\cdots\!00\)\( - \)\(50\!\cdots\!75\)\( \beta_{1} + \)\(29\!\cdots\!00\)\( \beta_{2}) q^{75} +(\)\(20\!\cdots\!00\)\( + \)\(83\!\cdots\!72\)\( \beta_{1} - \)\(28\!\cdots\!28\)\( \beta_{2}) q^{76} +(-\)\(10\!\cdots\!12\)\( - \)\(13\!\cdots\!20\)\( \beta_{1} + \)\(60\!\cdots\!84\)\( \beta_{2}) q^{77} +(\)\(41\!\cdots\!92\)\( + \)\(83\!\cdots\!88\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2}) q^{78} +(-\)\(25\!\cdots\!60\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} - \)\(28\!\cdots\!44\)\( \beta_{2}) q^{79} +(-\)\(12\!\cdots\!00\)\( - \)\(40\!\cdots\!20\)\( \beta_{1} + \)\(21\!\cdots\!60\)\( \beta_{2}) q^{80} +(-\)\(18\!\cdots\!19\)\( + \)\(46\!\cdots\!60\)\( \beta_{1} + \)\(75\!\cdots\!14\)\( \beta_{2}) q^{81} +(-\)\(16\!\cdots\!24\)\( + \)\(40\!\cdots\!48\)\( \beta_{1} - \)\(17\!\cdots\!92\)\( \beta_{2}) q^{82} +(-\)\(88\!\cdots\!52\)\( + \)\(36\!\cdots\!83\)\( \beta_{1} - \)\(91\!\cdots\!24\)\( \beta_{2}) q^{83} +(-\)\(43\!\cdots\!12\)\( - \)\(17\!\cdots\!56\)\( \beta_{1} + \)\(19\!\cdots\!52\)\( \beta_{2}) q^{84} +(-\)\(20\!\cdots\!00\)\( - \)\(20\!\cdots\!80\)\( \beta_{1} - \)\(46\!\cdots\!10\)\( \beta_{2}) q^{85} +(-\)\(17\!\cdots\!16\)\( - \)\(28\!\cdots\!96\)\( \beta_{1} - \)\(35\!\cdots\!44\)\( \beta_{2}) q^{86} +(-\)\(37\!\cdots\!20\)\( + \)\(31\!\cdots\!34\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2}) q^{87} +(-\)\(68\!\cdots\!16\)\( - \)\(99\!\cdots\!20\)\( \beta_{1} + \)\(88\!\cdots\!76\)\( \beta_{2}) q^{88} +(-\)\(99\!\cdots\!90\)\( + \)\(67\!\cdots\!20\)\( \beta_{1} + \)\(72\!\cdots\!50\)\( \beta_{2}) q^{89} +(-\)\(55\!\cdots\!00\)\( - \)\(39\!\cdots\!20\)\( \beta_{1} - \)\(93\!\cdots\!40\)\( \beta_{2}) q^{90} +(-\)\(10\!\cdots\!28\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(33\!\cdots\!24\)\( \beta_{2}) q^{91} +(\)\(83\!\cdots\!52\)\( + \)\(18\!\cdots\!00\)\( \beta_{1} - \)\(65\!\cdots\!52\)\( \beta_{2}) q^{92} +(\)\(67\!\cdots\!16\)\( + \)\(73\!\cdots\!08\)\( \beta_{1} - \)\(21\!\cdots\!52\)\( \beta_{2}) q^{93} +(-\)\(21\!\cdots\!28\)\( + \)\(18\!\cdots\!20\)\( \beta_{1} + \)\(37\!\cdots\!68\)\( \beta_{2}) q^{94} +(-\)\(92\!\cdots\!00\)\( - \)\(16\!\cdots\!50\)\( \beta_{1} + \)\(16\!\cdots\!00\)\( \beta_{2}) q^{95} +(-\)\(76\!\cdots\!16\)\( - \)\(50\!\cdots\!68\)\( \beta_{1}) q^{96} +(-\)\(11\!\cdots\!66\)\( - \)\(29\!\cdots\!52\)\( \beta_{1} - \)\(21\!\cdots\!14\)\( \beta_{2}) q^{97} +(\)\(17\!\cdots\!24\)\( + \)\(27\!\cdots\!32\)\( \beta_{1} - \)\(14\!\cdots\!32\)\( \beta_{2}) q^{98} +(\)\(14\!\cdots\!44\)\( - \)\(48\!\cdots\!63\)\( \beta_{1} + \)\(66\!\cdots\!64\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 1649267441664q^{2} - 4584969341748725436q^{3} + \)\(90\!\cdots\!32\)\(q^{4} - \)\(40\!\cdots\!50\)\(q^{5} - \)\(25\!\cdots\!68\)\(q^{6} + \)\(14\!\cdots\!12\)\(q^{7} + \)\(49\!\cdots\!16\)\(q^{8} - \)\(66\!\cdots\!69\)\(q^{9} + O(q^{10}) \) \( 3q + 1649267441664q^{2} - 4584969341748725436q^{3} + \)\(90\!\cdots\!32\)\(q^{4} - \)\(40\!\cdots\!50\)\(q^{5} - \)\(25\!\cdots\!68\)\(q^{6} + \)\(14\!\cdots\!12\)\(q^{7} + \)\(49\!\cdots\!16\)\(q^{8} - \)\(66\!\cdots\!69\)\(q^{9} - \)\(22\!\cdots\!00\)\(q^{10} - \)\(12\!\cdots\!84\)\(q^{11} - \)\(13\!\cdots\!84\)\(q^{12} + \)\(42\!\cdots\!54\)\(q^{13} + \)\(81\!\cdots\!56\)\(q^{14} + \)\(12\!\cdots\!00\)\(q^{15} + \)\(27\!\cdots\!08\)\(q^{16} + \)\(94\!\cdots\!82\)\(q^{17} - \)\(36\!\cdots\!72\)\(q^{18} + \)\(20\!\cdots\!00\)\(q^{19} - \)\(12\!\cdots\!00\)\(q^{20} - \)\(43\!\cdots\!44\)\(q^{21} - \)\(68\!\cdots\!92\)\(q^{22} + \)\(82\!\cdots\!24\)\(q^{23} - \)\(76\!\cdots\!92\)\(q^{24} + \)\(21\!\cdots\!25\)\(q^{25} + \)\(23\!\cdots\!52\)\(q^{26} + \)\(27\!\cdots\!40\)\(q^{27} + \)\(44\!\cdots\!28\)\(q^{28} + \)\(14\!\cdots\!30\)\(q^{29} + \)\(66\!\cdots\!00\)\(q^{30} - \)\(95\!\cdots\!04\)\(q^{31} + \)\(15\!\cdots\!04\)\(q^{32} + \)\(99\!\cdots\!08\)\(q^{33} + \)\(52\!\cdots\!16\)\(q^{34} - \)\(21\!\cdots\!00\)\(q^{35} - \)\(20\!\cdots\!36\)\(q^{36} - \)\(14\!\cdots\!58\)\(q^{37} + \)\(11\!\cdots\!00\)\(q^{38} + \)\(22\!\cdots\!52\)\(q^{39} - \)\(67\!\cdots\!00\)\(q^{40} - \)\(89\!\cdots\!94\)\(q^{41} - \)\(23\!\cdots\!72\)\(q^{42} - \)\(95\!\cdots\!96\)\(q^{43} - \)\(37\!\cdots\!96\)\(q^{44} - \)\(30\!\cdots\!50\)\(q^{45} + \)\(45\!\cdots\!12\)\(q^{46} - \)\(11\!\cdots\!68\)\(q^{47} - \)\(41\!\cdots\!96\)\(q^{48} + \)\(97\!\cdots\!19\)\(q^{49} + \)\(11\!\cdots\!00\)\(q^{50} + \)\(54\!\cdots\!16\)\(q^{51} + \)\(12\!\cdots\!76\)\(q^{52} + \)\(12\!\cdots\!74\)\(q^{53} + \)\(15\!\cdots\!20\)\(q^{54} + \)\(21\!\cdots\!00\)\(q^{55} + \)\(24\!\cdots\!64\)\(q^{56} - \)\(37\!\cdots\!00\)\(q^{57} + \)\(82\!\cdots\!40\)\(q^{58} - \)\(11\!\cdots\!40\)\(q^{59} + \)\(36\!\cdots\!00\)\(q^{60} - \)\(64\!\cdots\!14\)\(q^{61} - \)\(52\!\cdots\!52\)\(q^{62} + \)\(70\!\cdots\!24\)\(q^{63} + \)\(82\!\cdots\!52\)\(q^{64} + \)\(89\!\cdots\!00\)\(q^{65} + \)\(54\!\cdots\!04\)\(q^{66} + \)\(10\!\cdots\!52\)\(q^{67} + \)\(28\!\cdots\!08\)\(q^{68} - \)\(81\!\cdots\!88\)\(q^{69} - \)\(11\!\cdots\!00\)\(q^{70} - \)\(11\!\cdots\!44\)\(q^{71} - \)\(11\!\cdots\!68\)\(q^{72} - \)\(21\!\cdots\!66\)\(q^{73} - \)\(79\!\cdots\!04\)\(q^{74} - \)\(29\!\cdots\!00\)\(q^{75} + \)\(61\!\cdots\!00\)\(q^{76} - \)\(31\!\cdots\!36\)\(q^{77} + \)\(12\!\cdots\!76\)\(q^{78} - \)\(75\!\cdots\!80\)\(q^{79} - \)\(37\!\cdots\!00\)\(q^{80} - \)\(54\!\cdots\!57\)\(q^{81} - \)\(49\!\cdots\!72\)\(q^{82} - \)\(26\!\cdots\!56\)\(q^{83} - \)\(13\!\cdots\!36\)\(q^{84} - \)\(61\!\cdots\!00\)\(q^{85} - \)\(52\!\cdots\!48\)\(q^{86} - \)\(11\!\cdots\!60\)\(q^{87} - \)\(20\!\cdots\!48\)\(q^{88} - \)\(29\!\cdots\!70\)\(q^{89} - \)\(16\!\cdots\!00\)\(q^{90} - \)\(32\!\cdots\!84\)\(q^{91} + \)\(24\!\cdots\!56\)\(q^{92} + \)\(20\!\cdots\!48\)\(q^{93} - \)\(65\!\cdots\!84\)\(q^{94} - \)\(27\!\cdots\!00\)\(q^{95} - \)\(23\!\cdots\!48\)\(q^{96} - \)\(35\!\cdots\!98\)\(q^{97} + \)\(53\!\cdots\!72\)\(q^{98} + \)\(43\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5157021731103247543589585180 x + 141562397820564875200991893221092433132672\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 288 \nu^{2} + 10566252004493664 \nu - 990148172371827050453201852544 \)\()/ 16696982899 \)
\(\beta_{2}\)\(=\)\((\)\( 788207616 \nu^{2} + 36596022868804296296448 \nu - 2709869202888734729468024946465988608 \)\()/ 16696982899 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2736832 \beta_{1} + 153280512000\)\()/ 459841536000 \)
\(\nu^{2}\)\(=\)\((\)\(-36688375015603 \beta_{2} + 127069523850014917696 \beta_{1} + 1580941862677264216688307868669304832000\)\()/ 459841536000 \)

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
−8.28584e13
4.42382e13
3.86202e13
5.49756e11 −8.21312e18 3.02231e23 −6.30299e27 −4.51521e30 3.82944e33 1.66153e35 1.81857e37 −3.46510e39
1.2 5.49756e11 −3.97815e18 3.02231e23 4.89479e27 −2.18701e30 −4.89031e32 1.66153e35 −3.34439e37 2.69094e39
1.3 5.49756e11 7.60629e18 3.02231e23 −2.65106e27 4.18160e30 −1.85699e33 1.66153e35 8.58611e36 −1.45744e39
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2.80.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.80.a.a 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + \)\(45\!\cdots\!36\)\( T_{3}^{2} - \)\(60\!\cdots\!68\)\( T_{3} - \)\(24\!\cdots\!72\)\( \) acting on \(S_{80}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 549755813888 T )^{3} \)
$3$ \( 1 + 4584969341748725436 T + \)\(87\!\cdots\!33\)\( T^{2} + \)\(20\!\cdots\!52\)\( T^{3} + \)\(43\!\cdots\!11\)\( T^{4} + \)\(11\!\cdots\!04\)\( T^{5} + \)\(11\!\cdots\!63\)\( T^{6} \)
$5$ \( 1 + \)\(40\!\cdots\!50\)\( T + \)\(22\!\cdots\!75\)\( T^{2} + \)\(52\!\cdots\!00\)\( T^{3} + \)\(37\!\cdots\!75\)\( T^{4} + \)\(11\!\cdots\!50\)\( T^{5} + \)\(45\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - \)\(14\!\cdots\!12\)\( T + \)\(92\!\cdots\!77\)\( T^{2} - \)\(20\!\cdots\!96\)\( T^{3} + \)\(53\!\cdots\!11\)\( T^{4} - \)\(49\!\cdots\!88\)\( T^{5} + \)\(19\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 + \)\(12\!\cdots\!84\)\( T + \)\(58\!\cdots\!25\)\( T^{2} + \)\(45\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} + \)\(43\!\cdots\!04\)\( T^{5} + \)\(64\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 - \)\(42\!\cdots\!54\)\( T + \)\(23\!\cdots\!03\)\( T^{2} - \)\(79\!\cdots\!48\)\( T^{3} + \)\(23\!\cdots\!31\)\( T^{4} - \)\(43\!\cdots\!66\)\( T^{5} + \)\(10\!\cdots\!33\)\( T^{6} \)
$17$ \( 1 - \)\(94\!\cdots\!82\)\( T + \)\(59\!\cdots\!67\)\( T^{2} + \)\(50\!\cdots\!24\)\( T^{3} + \)\(96\!\cdots\!51\)\( T^{4} - \)\(24\!\cdots\!38\)\( T^{5} + \)\(41\!\cdots\!77\)\( T^{6} \)
$19$ \( 1 - \)\(20\!\cdots\!00\)\( T + \)\(21\!\cdots\!37\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!23\)\( T^{4} - \)\(22\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 - \)\(82\!\cdots\!24\)\( T + \)\(62\!\cdots\!53\)\( T^{2} - \)\(18\!\cdots\!88\)\( T^{3} + \)\(23\!\cdots\!11\)\( T^{4} - \)\(11\!\cdots\!56\)\( T^{5} + \)\(53\!\cdots\!03\)\( T^{6} \)
$29$ \( 1 - \)\(14\!\cdots\!30\)\( T + \)\(52\!\cdots\!07\)\( T^{2} - \)\(70\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!83\)\( T^{4} - \)\(17\!\cdots\!30\)\( T^{5} + \)\(38\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 + \)\(95\!\cdots\!04\)\( T + \)\(11\!\cdots\!85\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{3} + \)\(78\!\cdots\!35\)\( T^{4} + \)\(41\!\cdots\!64\)\( T^{5} + \)\(28\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 + \)\(14\!\cdots\!58\)\( T + \)\(24\!\cdots\!07\)\( T^{2} + \)\(19\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!11\)\( T^{4} + \)\(86\!\cdots\!82\)\( T^{5} + \)\(46\!\cdots\!17\)\( T^{6} \)
$41$ \( 1 + \)\(89\!\cdots\!94\)\( T + \)\(66\!\cdots\!95\)\( T^{2} + \)\(41\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!95\)\( T^{4} + \)\(59\!\cdots\!74\)\( T^{5} + \)\(16\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 + \)\(95\!\cdots\!96\)\( T + \)\(43\!\cdots\!93\)\( T^{2} + \)\(14\!\cdots\!12\)\( T^{3} + \)\(47\!\cdots\!51\)\( T^{4} + \)\(11\!\cdots\!04\)\( T^{5} + \)\(13\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 + \)\(11\!\cdots\!68\)\( T + \)\(14\!\cdots\!57\)\( T^{2} + \)\(26\!\cdots\!04\)\( T^{3} + \)\(18\!\cdots\!31\)\( T^{4} + \)\(18\!\cdots\!52\)\( T^{5} + \)\(19\!\cdots\!87\)\( T^{6} \)
$53$ \( 1 - \)\(12\!\cdots\!74\)\( T + \)\(29\!\cdots\!43\)\( T^{2} - \)\(46\!\cdots\!28\)\( T^{3} + \)\(49\!\cdots\!31\)\( T^{4} - \)\(34\!\cdots\!86\)\( T^{5} + \)\(45\!\cdots\!13\)\( T^{6} \)
$59$ \( 1 + \)\(11\!\cdots\!40\)\( T + \)\(20\!\cdots\!17\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!63\)\( T^{4} + \)\(74\!\cdots\!40\)\( T^{5} + \)\(49\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 + \)\(64\!\cdots\!14\)\( T + \)\(27\!\cdots\!55\)\( T^{2} + \)\(92\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!55\)\( T^{4} + \)\(77\!\cdots\!34\)\( T^{5} + \)\(13\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 - \)\(10\!\cdots\!52\)\( T + \)\(56\!\cdots\!77\)\( T^{2} - \)\(38\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} - \)\(35\!\cdots\!68\)\( T^{5} + \)\(60\!\cdots\!27\)\( T^{6} \)
$71$ \( 1 + \)\(11\!\cdots\!44\)\( T + \)\(47\!\cdots\!05\)\( T^{2} + \)\(34\!\cdots\!20\)\( T^{3} + \)\(83\!\cdots\!55\)\( T^{4} + \)\(37\!\cdots\!84\)\( T^{5} + \)\(56\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 + \)\(21\!\cdots\!66\)\( T + \)\(30\!\cdots\!63\)\( T^{2} + \)\(24\!\cdots\!32\)\( T^{3} + \)\(48\!\cdots\!31\)\( T^{4} + \)\(53\!\cdots\!54\)\( T^{5} + \)\(40\!\cdots\!53\)\( T^{6} \)
$79$ \( 1 + \)\(75\!\cdots\!80\)\( T + \)\(14\!\cdots\!57\)\( T^{2} + \)\(77\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} + \)\(50\!\cdots\!80\)\( T^{5} + \)\(54\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 + \)\(26\!\cdots\!56\)\( T + \)\(34\!\cdots\!53\)\( T^{2} + \)\(27\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!91\)\( T^{4} + \)\(43\!\cdots\!04\)\( T^{5} + \)\(66\!\cdots\!23\)\( T^{6} \)
$89$ \( 1 + \)\(29\!\cdots\!70\)\( T + \)\(56\!\cdots\!27\)\( T^{2} + \)\(67\!\cdots\!60\)\( T^{3} + \)\(56\!\cdots\!43\)\( T^{4} + \)\(30\!\cdots\!70\)\( T^{5} + \)\(10\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 + \)\(35\!\cdots\!98\)\( T + \)\(25\!\cdots\!67\)\( T^{2} + \)\(55\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!11\)\( T^{4} + \)\(29\!\cdots\!22\)\( T^{5} + \)\(73\!\cdots\!37\)\( T^{6} \)
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