Properties

Label 2.80.a.b
Level 2
Weight 80
Character orbit 2.a
Self dual yes
Analytic conductor 79.047
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{12}\cdot 5^{5}\cdot 7\cdot 11\cdot 13 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -549755813888 q^{2} +(1099291558848636972 - \beta_{1}) q^{3} +\)\(30\!\cdots\!44\)\( q^{4} +(-\)\(52\!\cdots\!70\)\( + 187080110 \beta_{1} - \beta_{2}) q^{5} +(-\)\(60\!\cdots\!36\)\( + 549755813888 \beta_{1}) q^{6} +(\)\(64\!\cdots\!76\)\( - 139800155710429 \beta_{1} - 25291 \beta_{2} + 27 \beta_{3}) q^{7} -\)\(16\!\cdots\!72\)\( q^{8} +(\)\(28\!\cdots\!17\)\( - 601742752495371368 \beta_{1} + 19560781326 \beta_{2} + 252028 \beta_{3}) q^{9} +O(q^{10})\) \( q -549755813888 q^{2} +(1099291558848636972 - \beta_{1}) q^{3} +\)\(30\!\cdots\!44\)\( q^{4} +(-\)\(52\!\cdots\!70\)\( + 187080110 \beta_{1} - \beta_{2}) q^{5} +(-\)\(60\!\cdots\!36\)\( + 549755813888 \beta_{1}) q^{6} +(\)\(64\!\cdots\!76\)\( - 139800155710429 \beta_{1} - 25291 \beta_{2} + 27 \beta_{3}) q^{7} -\)\(16\!\cdots\!72\)\( q^{8} +(\)\(28\!\cdots\!17\)\( - 601742752495371368 \beta_{1} + 19560781326 \beta_{2} + 252028 \beta_{3}) q^{9} +(\)\(28\!\cdots\!60\)\( - \)\(10\!\cdots\!80\)\( \beta_{1} + 549755813888 \beta_{2}) q^{10} +(-\)\(49\!\cdots\!68\)\( + \)\(39\!\cdots\!75\)\( \beta_{1} - 25812417139174 \beta_{2} + 1098449478 \beta_{3}) q^{11} +(\)\(33\!\cdots\!68\)\( - \)\(30\!\cdots\!44\)\( \beta_{1}) q^{12} +(\)\(42\!\cdots\!02\)\( + \)\(59\!\cdots\!22\)\( \beta_{1} + 3172061340864439 \beta_{2} + 1387993835592 \beta_{3}) q^{13} +(-\)\(35\!\cdots\!88\)\( + \)\(76\!\cdots\!52\)\( \beta_{1} + 13903874289041408 \beta_{2} - 14843406974976 \beta_{3}) q^{14} +(-\)\(14\!\cdots\!40\)\( + \)\(38\!\cdots\!45\)\( \beta_{1} - 5303978607792266397 \beta_{2} - 78487606141875 \beta_{3}) q^{15} +\)\(91\!\cdots\!36\)\( q^{16} +(\)\(69\!\cdots\!26\)\( - \)\(15\!\cdots\!40\)\( \beta_{1} - \)\(76\!\cdots\!98\)\( \beta_{2} + 24096981171375756 \beta_{3}) q^{17} +(-\)\(15\!\cdots\!96\)\( + \)\(33\!\cdots\!84\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2} - 138553858262564864 \beta_{3}) q^{18} +(\)\(17\!\cdots\!40\)\( + \)\(14\!\cdots\!57\)\( \beta_{1} + \)\(67\!\cdots\!66\)\( \beta_{2} - 506911584947427702 \beta_{3}) q^{19} +(-\)\(15\!\cdots\!80\)\( + \)\(56\!\cdots\!40\)\( \beta_{1} - \)\(30\!\cdots\!44\)\( \beta_{2}) q^{20} +(\)\(11\!\cdots\!72\)\( - \)\(12\!\cdots\!76\)\( \beta_{1} + \)\(32\!\cdots\!36\)\( \beta_{2} + 92514685654531367208 \beta_{3}) q^{21} +(\)\(27\!\cdots\!84\)\( - \)\(21\!\cdots\!00\)\( \beta_{1} + \)\(14\!\cdots\!12\)\( \beta_{2} - \)\(60\!\cdots\!64\)\( \beta_{3}) q^{22} +(\)\(12\!\cdots\!52\)\( - \)\(35\!\cdots\!75\)\( \beta_{1} - \)\(16\!\cdots\!89\)\( \beta_{2} + \)\(49\!\cdots\!33\)\( \beta_{3}) q^{23} +(-\)\(18\!\cdots\!84\)\( + \)\(16\!\cdots\!72\)\( \beta_{1}) q^{24} +(-\)\(28\!\cdots\!25\)\( - \)\(15\!\cdots\!00\)\( \beta_{1} + \)\(90\!\cdots\!80\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3}) q^{25} +(-\)\(23\!\cdots\!76\)\( - \)\(32\!\cdots\!36\)\( \beta_{1} - \)\(17\!\cdots\!32\)\( \beta_{2} - \)\(76\!\cdots\!96\)\( \beta_{3}) q^{26} +(\)\(22\!\cdots\!00\)\( - \)\(52\!\cdots\!52\)\( \beta_{1} + \)\(48\!\cdots\!66\)\( \beta_{2} + \)\(13\!\cdots\!98\)\( \beta_{3}) q^{27} +(\)\(19\!\cdots\!44\)\( - \)\(42\!\cdots\!76\)\( \beta_{1} - \)\(76\!\cdots\!04\)\( \beta_{2} + \)\(81\!\cdots\!88\)\( \beta_{3}) q^{28} +(\)\(92\!\cdots\!90\)\( - \)\(44\!\cdots\!62\)\( \beta_{1} - \)\(15\!\cdots\!09\)\( \beta_{2} - \)\(42\!\cdots\!52\)\( \beta_{3}) q^{29} +(\)\(81\!\cdots\!20\)\( - \)\(21\!\cdots\!60\)\( \beta_{1} + \)\(29\!\cdots\!36\)\( \beta_{2} + \)\(43\!\cdots\!00\)\( \beta_{3}) q^{30} +(\)\(11\!\cdots\!52\)\( - \)\(83\!\cdots\!56\)\( \beta_{1} + \)\(73\!\cdots\!36\)\( \beta_{2} + \)\(10\!\cdots\!08\)\( \beta_{3}) q^{31} -\)\(50\!\cdots\!68\)\( q^{32} +(-\)\(35\!\cdots\!96\)\( + \)\(10\!\cdots\!44\)\( \beta_{1} - \)\(99\!\cdots\!02\)\( \beta_{2} + \)\(54\!\cdots\!44\)\( \beta_{3}) q^{33} +(-\)\(38\!\cdots\!88\)\( + \)\(83\!\cdots\!20\)\( \beta_{1} + \)\(42\!\cdots\!24\)\( \beta_{2} - \)\(13\!\cdots\!28\)\( \beta_{3}) q^{34} +(-\)\(19\!\cdots\!20\)\( + \)\(83\!\cdots\!60\)\( \beta_{1} - \)\(26\!\cdots\!76\)\( \beta_{2} + \)\(36\!\cdots\!00\)\( \beta_{3}) q^{35} +(\)\(85\!\cdots\!48\)\( - \)\(18\!\cdots\!92\)\( \beta_{1} + \)\(59\!\cdots\!44\)\( \beta_{2} + \)\(76\!\cdots\!32\)\( \beta_{3}) q^{36} +(-\)\(96\!\cdots\!94\)\( + \)\(56\!\cdots\!30\)\( \beta_{1} + \)\(12\!\cdots\!47\)\( \beta_{2} - \)\(48\!\cdots\!84\)\( \beta_{3}) q^{37} +(-\)\(94\!\cdots\!20\)\( - \)\(82\!\cdots\!16\)\( \beta_{1} - \)\(36\!\cdots\!08\)\( \beta_{2} + \)\(27\!\cdots\!76\)\( \beta_{3}) q^{38} +(-\)\(40\!\cdots\!56\)\( - \)\(94\!\cdots\!51\)\( \beta_{1} - \)\(84\!\cdots\!25\)\( \beta_{2} + \)\(15\!\cdots\!25\)\( \beta_{3}) q^{39} +(\)\(86\!\cdots\!40\)\( - \)\(31\!\cdots\!20\)\( \beta_{1} + \)\(16\!\cdots\!72\)\( \beta_{2}) q^{40} +(-\)\(31\!\cdots\!18\)\( - \)\(14\!\cdots\!96\)\( \beta_{1} + \)\(51\!\cdots\!72\)\( \beta_{2} - \)\(22\!\cdots\!84\)\( \beta_{3}) q^{41} +(-\)\(62\!\cdots\!36\)\( + \)\(67\!\cdots\!88\)\( \beta_{1} - \)\(17\!\cdots\!68\)\( \beta_{2} - \)\(50\!\cdots\!04\)\( \beta_{3}) q^{42} +(-\)\(69\!\cdots\!88\)\( + \)\(69\!\cdots\!53\)\( \beta_{1} + \)\(56\!\cdots\!16\)\( \beta_{2} + \)\(11\!\cdots\!48\)\( \beta_{3}) q^{43} +(-\)\(14\!\cdots\!92\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(78\!\cdots\!56\)\( \beta_{2} + \)\(33\!\cdots\!32\)\( \beta_{3}) q^{44} +(-\)\(28\!\cdots\!90\)\( + \)\(23\!\cdots\!70\)\( \beta_{1} - \)\(36\!\cdots\!17\)\( \beta_{2} - \)\(13\!\cdots\!00\)\( \beta_{3}) q^{45} +(-\)\(67\!\cdots\!76\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} + \)\(89\!\cdots\!32\)\( \beta_{2} - \)\(27\!\cdots\!04\)\( \beta_{3}) q^{46} +(\)\(14\!\cdots\!36\)\( - \)\(13\!\cdots\!30\)\( \beta_{1} + \)\(55\!\cdots\!18\)\( \beta_{2} + \)\(49\!\cdots\!54\)\( \beta_{3}) q^{47} +(\)\(10\!\cdots\!92\)\( - \)\(91\!\cdots\!36\)\( \beta_{1}) q^{48} +(\)\(21\!\cdots\!33\)\( - \)\(40\!\cdots\!64\)\( \beta_{1} - \)\(65\!\cdots\!88\)\( \beta_{2} - \)\(51\!\cdots\!64\)\( \beta_{3}) q^{49} +(\)\(15\!\cdots\!00\)\( + \)\(83\!\cdots\!00\)\( \beta_{1} - \)\(49\!\cdots\!40\)\( \beta_{2} - \)\(55\!\cdots\!00\)\( \beta_{3}) q^{50} +(\)\(12\!\cdots\!72\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} + \)\(21\!\cdots\!86\)\( \beta_{2} + \)\(65\!\cdots\!58\)\( \beta_{3}) q^{51} +(\)\(12\!\cdots\!88\)\( + \)\(18\!\cdots\!68\)\( \beta_{1} + \)\(95\!\cdots\!16\)\( \beta_{2} + \)\(41\!\cdots\!48\)\( \beta_{3}) q^{52} +(\)\(26\!\cdots\!62\)\( - \)\(52\!\cdots\!74\)\( \beta_{1} + \)\(91\!\cdots\!23\)\( \beta_{2} - \)\(94\!\cdots\!56\)\( \beta_{3}) q^{53} +(-\)\(12\!\cdots\!00\)\( + \)\(28\!\cdots\!76\)\( \beta_{1} - \)\(26\!\cdots\!08\)\( \beta_{2} - \)\(71\!\cdots\!24\)\( \beta_{3}) q^{54} +(\)\(42\!\cdots\!60\)\( - \)\(33\!\cdots\!05\)\( \beta_{1} - \)\(25\!\cdots\!07\)\( \beta_{2} + \)\(43\!\cdots\!75\)\( \beta_{3}) q^{55} +(-\)\(10\!\cdots\!72\)\( + \)\(23\!\cdots\!88\)\( \beta_{1} + \)\(42\!\cdots\!52\)\( \beta_{2} - \)\(44\!\cdots\!44\)\( \beta_{3}) q^{56} +(-\)\(95\!\cdots\!20\)\( - \)\(39\!\cdots\!52\)\( \beta_{1} - \)\(19\!\cdots\!14\)\( \beta_{2} - \)\(27\!\cdots\!92\)\( \beta_{3}) q^{57} +(-\)\(50\!\cdots\!20\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} + \)\(83\!\cdots\!92\)\( \beta_{2} + \)\(23\!\cdots\!76\)\( \beta_{3}) q^{58} +(-\)\(57\!\cdots\!20\)\( - \)\(10\!\cdots\!19\)\( \beta_{1} + \)\(12\!\cdots\!92\)\( \beta_{2} - \)\(51\!\cdots\!24\)\( \beta_{3}) q^{59} +(-\)\(44\!\cdots\!60\)\( + \)\(11\!\cdots\!80\)\( \beta_{1} - \)\(16\!\cdots\!68\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3}) q^{60} +(-\)\(59\!\cdots\!98\)\( + \)\(23\!\cdots\!78\)\( \beta_{1} - \)\(28\!\cdots\!33\)\( \beta_{2} + \)\(24\!\cdots\!76\)\( \beta_{3}) q^{61} +(-\)\(65\!\cdots\!76\)\( + \)\(45\!\cdots\!28\)\( \beta_{1} - \)\(40\!\cdots\!68\)\( \beta_{2} - \)\(58\!\cdots\!04\)\( \beta_{3}) q^{62} +(\)\(74\!\cdots\!92\)\( - \)\(17\!\cdots\!89\)\( \beta_{1} + \)\(32\!\cdots\!01\)\( \beta_{2} - \)\(72\!\cdots\!97\)\( \beta_{3}) q^{63} +\)\(27\!\cdots\!84\)\( q^{64} +(\)\(20\!\cdots\!60\)\( + \)\(38\!\cdots\!20\)\( \beta_{1} - \)\(89\!\cdots\!52\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3}) q^{65} +(\)\(19\!\cdots\!48\)\( - \)\(58\!\cdots\!72\)\( \beta_{1} + \)\(54\!\cdots\!76\)\( \beta_{2} - \)\(30\!\cdots\!72\)\( \beta_{3}) q^{66} +(\)\(74\!\cdots\!56\)\( + \)\(80\!\cdots\!89\)\( \beta_{1} + \)\(99\!\cdots\!98\)\( \beta_{2} - \)\(68\!\cdots\!06\)\( \beta_{3}) q^{67} +(\)\(21\!\cdots\!44\)\( - \)\(45\!\cdots\!60\)\( \beta_{1} - \)\(23\!\cdots\!12\)\( \beta_{2} + \)\(72\!\cdots\!64\)\( \beta_{3}) q^{68} +(\)\(16\!\cdots\!44\)\( + \)\(42\!\cdots\!84\)\( \beta_{1} - \)\(25\!\cdots\!72\)\( \beta_{2} - \)\(39\!\cdots\!16\)\( \beta_{3}) q^{69} +(\)\(10\!\cdots\!60\)\( - \)\(46\!\cdots\!80\)\( \beta_{1} + \)\(14\!\cdots\!88\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{70} +(-\)\(41\!\cdots\!08\)\( + \)\(11\!\cdots\!87\)\( \beta_{1} + \)\(44\!\cdots\!41\)\( \beta_{2} + \)\(10\!\cdots\!23\)\( \beta_{3}) q^{71} +(-\)\(46\!\cdots\!24\)\( + \)\(99\!\cdots\!96\)\( \beta_{1} - \)\(32\!\cdots\!72\)\( \beta_{2} - \)\(41\!\cdots\!16\)\( \beta_{3}) q^{72} +(-\)\(12\!\cdots\!58\)\( - \)\(35\!\cdots\!24\)\( \beta_{1} - \)\(25\!\cdots\!02\)\( \beta_{2} - \)\(34\!\cdots\!56\)\( \beta_{3}) q^{73} +(\)\(52\!\cdots\!72\)\( - \)\(30\!\cdots\!40\)\( \beta_{1} - \)\(66\!\cdots\!36\)\( \beta_{2} + \)\(26\!\cdots\!92\)\( \beta_{3}) q^{74} +(\)\(11\!\cdots\!00\)\( - \)\(47\!\cdots\!75\)\( \beta_{1} + \)\(33\!\cdots\!60\)\( \beta_{2} + \)\(62\!\cdots\!00\)\( \beta_{3}) q^{75} +(\)\(51\!\cdots\!60\)\( + \)\(45\!\cdots\!08\)\( \beta_{1} + \)\(20\!\cdots\!04\)\( \beta_{2} - \)\(15\!\cdots\!88\)\( \beta_{3}) q^{76} +(\)\(17\!\cdots\!32\)\( + \)\(18\!\cdots\!60\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} - \)\(13\!\cdots\!44\)\( \beta_{3}) q^{77} +(\)\(22\!\cdots\!28\)\( + \)\(51\!\cdots\!88\)\( \beta_{1} + \)\(46\!\cdots\!00\)\( \beta_{2} - \)\(87\!\cdots\!00\)\( \beta_{3}) q^{78} +(\)\(94\!\cdots\!80\)\( - \)\(28\!\cdots\!42\)\( \beta_{1} + \)\(45\!\cdots\!22\)\( \beta_{2} + \)\(27\!\cdots\!66\)\( \beta_{3}) q^{79} +(-\)\(47\!\cdots\!20\)\( + \)\(17\!\cdots\!60\)\( \beta_{1} - \)\(91\!\cdots\!36\)\( \beta_{2}) q^{80} +(\)\(26\!\cdots\!61\)\( - \)\(16\!\cdots\!20\)\( \beta_{1} + \)\(16\!\cdots\!78\)\( \beta_{2} + \)\(51\!\cdots\!84\)\( \beta_{3}) q^{81} +(\)\(17\!\cdots\!84\)\( + \)\(78\!\cdots\!48\)\( \beta_{1} - \)\(28\!\cdots\!36\)\( \beta_{2} + \)\(12\!\cdots\!92\)\( \beta_{3}) q^{82} +(\)\(48\!\cdots\!52\)\( - \)\(29\!\cdots\!17\)\( \beta_{1} + \)\(59\!\cdots\!08\)\( \beta_{2} - \)\(47\!\cdots\!76\)\( \beta_{3}) q^{83} +(\)\(34\!\cdots\!68\)\( - \)\(36\!\cdots\!44\)\( \beta_{1} + \)\(97\!\cdots\!84\)\( \beta_{2} + \)\(27\!\cdots\!52\)\( \beta_{3}) q^{84} +(\)\(76\!\cdots\!80\)\( + \)\(32\!\cdots\!60\)\( \beta_{1} - \)\(29\!\cdots\!26\)\( \beta_{2} + \)\(93\!\cdots\!00\)\( \beta_{3}) q^{85} +(\)\(38\!\cdots\!44\)\( - \)\(38\!\cdots\!64\)\( \beta_{1} - \)\(30\!\cdots\!08\)\( \beta_{2} - \)\(64\!\cdots\!24\)\( \beta_{3}) q^{86} +(\)\(34\!\cdots\!80\)\( + \)\(56\!\cdots\!49\)\( \beta_{1} + \)\(53\!\cdots\!55\)\( \beta_{2} - \)\(28\!\cdots\!35\)\( \beta_{3}) q^{87} +(\)\(81\!\cdots\!96\)\( - \)\(66\!\cdots\!00\)\( \beta_{1} + \)\(42\!\cdots\!28\)\( \beta_{2} - \)\(18\!\cdots\!16\)\( \beta_{3}) q^{88} +(\)\(11\!\cdots\!10\)\( + \)\(32\!\cdots\!40\)\( \beta_{1} + \)\(18\!\cdots\!10\)\( \beta_{2} + \)\(97\!\cdots\!80\)\( \beta_{3}) q^{89} +(\)\(15\!\cdots\!20\)\( - \)\(13\!\cdots\!60\)\( \beta_{1} + \)\(19\!\cdots\!96\)\( \beta_{2} + \)\(71\!\cdots\!00\)\( \beta_{3}) q^{90} +(\)\(27\!\cdots\!52\)\( - \)\(10\!\cdots\!40\)\( \beta_{1} - \)\(67\!\cdots\!72\)\( \beta_{2} - \)\(11\!\cdots\!16\)\( \beta_{3}) q^{91} +(\)\(36\!\cdots\!88\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} - \)\(49\!\cdots\!16\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3}) q^{92} +(\)\(65\!\cdots\!44\)\( - \)\(35\!\cdots\!92\)\( \beta_{1} + \)\(17\!\cdots\!84\)\( \beta_{2} + \)\(25\!\cdots\!52\)\( \beta_{3}) q^{93} +(-\)\(79\!\cdots\!68\)\( + \)\(73\!\cdots\!40\)\( \beta_{1} - \)\(30\!\cdots\!84\)\( \beta_{2} - \)\(27\!\cdots\!52\)\( \beta_{3}) q^{94} +(-\)\(76\!\cdots\!00\)\( + \)\(22\!\cdots\!25\)\( \beta_{1} - \)\(54\!\cdots\!65\)\( \beta_{2} - \)\(54\!\cdots\!75\)\( \beta_{3}) q^{95} +(-\)\(55\!\cdots\!96\)\( + \)\(50\!\cdots\!68\)\( \beta_{1}) q^{96} +(\)\(21\!\cdots\!66\)\( - \)\(13\!\cdots\!92\)\( \beta_{1} - \)\(54\!\cdots\!02\)\( \beta_{2} + \)\(27\!\cdots\!44\)\( \beta_{3}) q^{97} +(-\)\(11\!\cdots\!04\)\( + \)\(22\!\cdots\!32\)\( \beta_{1} + \)\(35\!\cdots\!44\)\( \beta_{2} + \)\(28\!\cdots\!32\)\( \beta_{3}) q^{98} +(-\)\(60\!\cdots\!56\)\( + \)\(43\!\cdots\!93\)\( \beta_{1} - \)\(94\!\cdots\!32\)\( \beta_{2} - \)\(82\!\cdots\!96\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2199023255552q^{2} + 4397166235394547888q^{3} + \)\(12\!\cdots\!76\)\(q^{4} - \)\(20\!\cdots\!80\)\(q^{5} - \)\(24\!\cdots\!44\)\(q^{6} + \)\(25\!\cdots\!04\)\(q^{7} - \)\(66\!\cdots\!88\)\(q^{8} + \)\(11\!\cdots\!68\)\(q^{9} + O(q^{10}) \) \( 4q - 2199023255552q^{2} + 4397166235394547888q^{3} + \)\(12\!\cdots\!76\)\(q^{4} - \)\(20\!\cdots\!80\)\(q^{5} - \)\(24\!\cdots\!44\)\(q^{6} + \)\(25\!\cdots\!04\)\(q^{7} - \)\(66\!\cdots\!88\)\(q^{8} + \)\(11\!\cdots\!68\)\(q^{9} + \)\(11\!\cdots\!40\)\(q^{10} - \)\(19\!\cdots\!72\)\(q^{11} + \)\(13\!\cdots\!72\)\(q^{12} + \)\(17\!\cdots\!08\)\(q^{13} - \)\(14\!\cdots\!52\)\(q^{14} - \)\(59\!\cdots\!60\)\(q^{15} + \)\(36\!\cdots\!44\)\(q^{16} + \)\(27\!\cdots\!04\)\(q^{17} - \)\(62\!\cdots\!84\)\(q^{18} + \)\(68\!\cdots\!60\)\(q^{19} - \)\(63\!\cdots\!20\)\(q^{20} + \)\(45\!\cdots\!88\)\(q^{21} + \)\(10\!\cdots\!36\)\(q^{22} + \)\(48\!\cdots\!08\)\(q^{23} - \)\(73\!\cdots\!36\)\(q^{24} - \)\(11\!\cdots\!00\)\(q^{25} - \)\(93\!\cdots\!04\)\(q^{26} + \)\(90\!\cdots\!00\)\(q^{27} + \)\(77\!\cdots\!76\)\(q^{28} + \)\(36\!\cdots\!60\)\(q^{29} + \)\(32\!\cdots\!80\)\(q^{30} + \)\(47\!\cdots\!08\)\(q^{31} - \)\(20\!\cdots\!72\)\(q^{32} - \)\(14\!\cdots\!84\)\(q^{33} - \)\(15\!\cdots\!52\)\(q^{34} - \)\(79\!\cdots\!80\)\(q^{35} + \)\(34\!\cdots\!92\)\(q^{36} - \)\(38\!\cdots\!76\)\(q^{37} - \)\(37\!\cdots\!80\)\(q^{38} - \)\(16\!\cdots\!24\)\(q^{39} + \)\(34\!\cdots\!60\)\(q^{40} - \)\(12\!\cdots\!72\)\(q^{41} - \)\(25\!\cdots\!44\)\(q^{42} - \)\(27\!\cdots\!52\)\(q^{43} - \)\(59\!\cdots\!68\)\(q^{44} - \)\(11\!\cdots\!60\)\(q^{45} - \)\(26\!\cdots\!04\)\(q^{46} + \)\(57\!\cdots\!44\)\(q^{47} + \)\(40\!\cdots\!68\)\(q^{48} + \)\(85\!\cdots\!32\)\(q^{49} + \)\(61\!\cdots\!00\)\(q^{50} + \)\(49\!\cdots\!88\)\(q^{51} + \)\(51\!\cdots\!52\)\(q^{52} + \)\(10\!\cdots\!48\)\(q^{53} - \)\(50\!\cdots\!00\)\(q^{54} + \)\(17\!\cdots\!40\)\(q^{55} - \)\(42\!\cdots\!88\)\(q^{56} - \)\(38\!\cdots\!80\)\(q^{57} - \)\(20\!\cdots\!80\)\(q^{58} - \)\(22\!\cdots\!80\)\(q^{59} - \)\(17\!\cdots\!40\)\(q^{60} - \)\(23\!\cdots\!92\)\(q^{61} - \)\(26\!\cdots\!04\)\(q^{62} + \)\(29\!\cdots\!68\)\(q^{63} + \)\(11\!\cdots\!36\)\(q^{64} + \)\(82\!\cdots\!40\)\(q^{65} + \)\(78\!\cdots\!92\)\(q^{66} + \)\(29\!\cdots\!24\)\(q^{67} + \)\(84\!\cdots\!76\)\(q^{68} + \)\(64\!\cdots\!76\)\(q^{69} + \)\(43\!\cdots\!40\)\(q^{70} - \)\(16\!\cdots\!32\)\(q^{71} - \)\(18\!\cdots\!96\)\(q^{72} - \)\(51\!\cdots\!32\)\(q^{73} + \)\(21\!\cdots\!88\)\(q^{74} + \)\(46\!\cdots\!00\)\(q^{75} + \)\(20\!\cdots\!40\)\(q^{76} + \)\(71\!\cdots\!28\)\(q^{77} + \)\(89\!\cdots\!12\)\(q^{78} + \)\(37\!\cdots\!20\)\(q^{79} - \)\(19\!\cdots\!80\)\(q^{80} + \)\(10\!\cdots\!44\)\(q^{81} + \)\(69\!\cdots\!36\)\(q^{82} + \)\(19\!\cdots\!08\)\(q^{83} + \)\(13\!\cdots\!72\)\(q^{84} + \)\(30\!\cdots\!20\)\(q^{85} + \)\(15\!\cdots\!76\)\(q^{86} + \)\(13\!\cdots\!20\)\(q^{87} + \)\(32\!\cdots\!84\)\(q^{88} + \)\(45\!\cdots\!40\)\(q^{89} + \)\(62\!\cdots\!80\)\(q^{90} + \)\(10\!\cdots\!08\)\(q^{91} + \)\(14\!\cdots\!52\)\(q^{92} + \)\(26\!\cdots\!76\)\(q^{93} - \)\(31\!\cdots\!72\)\(q^{94} - \)\(30\!\cdots\!00\)\(q^{95} - \)\(22\!\cdots\!84\)\(q^{96} + \)\(87\!\cdots\!64\)\(q^{97} - \)\(46\!\cdots\!16\)\(q^{98} - \)\(24\!\cdots\!24\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 41378554124364514550149116389619 x^{2} - 22942689466335514146071616491770842136426264992 x + 189651623212039036292000549473742215529785444677917055760560\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 1920 \nu - 960 \)
\(\beta_{2}\)\(=\)\((\)\(94382325760 \nu^{3} + 92760753097633527527464960 \nu^{2} - 3981686980298358400765323209345000226236160 \nu - 3543191214600274264106051707090388311470836444850125397120\)\()/ \)\(70\!\cdots\!51\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-551370049454080 \nu^{3} + 230298538974084875840034283520 \nu^{2} + 22618304750182797236596274292018300663123822720 \nu + 4722723589396661633390365765156049711360059185095410142792640\)\()/ \)\(52\!\cdots\!49\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 960\)\()/1920\)
\(\nu^{2}\)\(=\)\((\)\(126014 \beta_{3} + 9780390663 \beta_{2} + 798420182600952248 \beta_{1} + 38134475481014336609417425664674713600\)\()/1843200\)
\(\nu^{3}\)\(=\)\((\)\(-15852667765708854802057 \beta_{3} + 522894631947995445632890341 \beta_{2} + 5083470331177408316416358829272845481 \beta_{1} + 4059644661537578132346348247197931573974100831646515200\)\()/ 235929600 \)

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
6.69343e15
8.14662e12
−5.66947e14
−6.13463e15
−5.49756e11 −1.17521e19 3.02231e23 −1.34628e27 6.46078e30 −6.77209e32 −1.66153e35 8.88420e37 7.40123e38
1.2 −5.49756e11 1.08365e18 3.02231e23 4.57934e27 −5.95743e29 3.27977e33 −1.66153e35 −4.80953e37 −2.51752e39
1.3 −5.49756e11 2.18783e18 3.02231e23 1.09007e27 −1.20277e30 −3.21221e33 −1.66153e35 −4.44830e37 −5.99271e38
1.4 −5.49756e11 1.28778e19 3.02231e23 −6.40915e27 −7.07963e30 3.18330e33 −1.66153e35 1.16567e38 3.52347e39
\(n\): 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.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.80.a.b 4 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 \( T_{3}^{4} - \)\(43\!\cdots\!88\)\( T_{3}^{3} - \)\(14\!\cdots\!96\)\( T_{3}^{2} + \)\(49\!\cdots\!08\)\( T_{3} - \)\(35\!\cdots\!44\)\( \) acting on \(S_{80}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 549755813888 T )^{4} \)
$3$ \( 1 - 4397166235394547888 T + \)\(51\!\cdots\!72\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} - \)\(11\!\cdots\!74\)\( T^{4} - \)\(77\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!08\)\( T^{6} - \)\(52\!\cdots\!44\)\( T^{7} + \)\(58\!\cdots\!21\)\( T^{8} \)
$5$ \( 1 + \)\(20\!\cdots\!80\)\( T + \)\(35\!\cdots\!00\)\( T^{2} + \)\(93\!\cdots\!00\)\( T^{3} + \)\(68\!\cdots\!50\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(98\!\cdots\!00\)\( T^{6} + \)\(94\!\cdots\!00\)\( T^{7} + \)\(74\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - \)\(25\!\cdots\!04\)\( T + \)\(10\!\cdots\!28\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(78\!\cdots\!86\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{5} + \)\(35\!\cdots\!72\)\( T^{6} - \)\(49\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 + \)\(19\!\cdots\!72\)\( T + \)\(49\!\cdots\!08\)\( T^{2} + \)\(42\!\cdots\!84\)\( T^{3} + \)\(85\!\cdots\!70\)\( T^{4} + \)\(79\!\cdots\!44\)\( T^{5} + \)\(16\!\cdots\!48\)\( T^{6} + \)\(12\!\cdots\!12\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$13$ \( 1 - \)\(17\!\cdots\!08\)\( T + \)\(13\!\cdots\!32\)\( T^{2} + \)\(33\!\cdots\!20\)\( T^{3} - \)\(80\!\cdots\!14\)\( T^{4} + \)\(33\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} - \)\(17\!\cdots\!64\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \)
$17$ \( 1 - \)\(27\!\cdots\!04\)\( T + \)\(38\!\cdots\!68\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(73\!\cdots\!66\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{5} + \)\(99\!\cdots\!12\)\( T^{6} - \)\(11\!\cdots\!08\)\( T^{7} + \)\(66\!\cdots\!81\)\( T^{8} \)
$19$ \( 1 - \)\(68\!\cdots\!60\)\( T + \)\(43\!\cdots\!16\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(66\!\cdots\!46\)\( T^{4} - \)\(19\!\cdots\!80\)\( T^{5} + \)\(48\!\cdots\!56\)\( T^{6} - \)\(79\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!81\)\( T^{8} \)
$23$ \( 1 - \)\(48\!\cdots\!08\)\( T + \)\(88\!\cdots\!72\)\( T^{2} - \)\(50\!\cdots\!20\)\( T^{3} + \)\(44\!\cdots\!06\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!68\)\( T^{6} - \)\(26\!\cdots\!24\)\( T^{7} + \)\(20\!\cdots\!61\)\( T^{8} \)
$29$ \( 1 - \)\(36\!\cdots\!60\)\( T + \)\(18\!\cdots\!76\)\( T^{2} + \)\(32\!\cdots\!80\)\( T^{3} - \)\(47\!\cdots\!34\)\( T^{4} + \)\(11\!\cdots\!20\)\( T^{5} + \)\(21\!\cdots\!36\)\( T^{6} - \)\(14\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!21\)\( T^{8} \)
$31$ \( 1 - \)\(47\!\cdots\!08\)\( T + \)\(14\!\cdots\!08\)\( T^{2} - \)\(11\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} - \)\(78\!\cdots\!56\)\( T^{5} + \)\(64\!\cdots\!28\)\( T^{6} - \)\(13\!\cdots\!88\)\( T^{7} + \)\(18\!\cdots\!81\)\( T^{8} \)
$37$ \( 1 + \)\(38\!\cdots\!76\)\( T + \)\(18\!\cdots\!08\)\( T^{2} + \)\(70\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!06\)\( T^{4} + \)\(54\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!92\)\( T^{7} + \)\(35\!\cdots\!41\)\( T^{8} \)
$41$ \( 1 + \)\(12\!\cdots\!72\)\( T + \)\(15\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!04\)\( T^{3} + \)\(64\!\cdots\!70\)\( T^{4} + \)\(26\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} + \)\(21\!\cdots\!32\)\( T^{7} + \)\(43\!\cdots\!41\)\( T^{8} \)
$43$ \( 1 + \)\(27\!\cdots\!52\)\( T + \)\(35\!\cdots\!92\)\( T^{2} + \)\(77\!\cdots\!80\)\( T^{3} + \)\(56\!\cdots\!26\)\( T^{4} + \)\(85\!\cdots\!60\)\( T^{5} + \)\(43\!\cdots\!08\)\( T^{6} + \)\(37\!\cdots\!36\)\( T^{7} + \)\(14\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(57\!\cdots\!44\)\( T + \)\(19\!\cdots\!08\)\( T^{2} - \)\(26\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!66\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{5} + \)\(29\!\cdots\!12\)\( T^{6} - \)\(11\!\cdots\!28\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 - \)\(10\!\cdots\!48\)\( T + \)\(49\!\cdots\!32\)\( T^{2} - \)\(41\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(68\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!48\)\( T^{6} - \)\(47\!\cdots\!24\)\( T^{7} + \)\(74\!\cdots\!21\)\( T^{8} \)
$59$ \( 1 + \)\(22\!\cdots\!80\)\( T + \)\(46\!\cdots\!56\)\( T^{2} + \)\(55\!\cdots\!60\)\( T^{3} + \)\(60\!\cdots\!26\)\( T^{4} + \)\(44\!\cdots\!40\)\( T^{5} + \)\(29\!\cdots\!76\)\( T^{6} + \)\(11\!\cdots\!20\)\( T^{7} + \)\(38\!\cdots\!41\)\( T^{8} \)
$61$ \( 1 + \)\(23\!\cdots\!92\)\( T + \)\(25\!\cdots\!88\)\( T^{2} + \)\(52\!\cdots\!84\)\( T^{3} + \)\(41\!\cdots\!70\)\( T^{4} + \)\(58\!\cdots\!44\)\( T^{5} + \)\(31\!\cdots\!28\)\( T^{6} + \)\(31\!\cdots\!32\)\( T^{7} + \)\(14\!\cdots\!61\)\( T^{8} \)
$67$ \( 1 - \)\(29\!\cdots\!24\)\( T + \)\(85\!\cdots\!28\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!46\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{5} + \)\(28\!\cdots\!52\)\( T^{6} - \)\(17\!\cdots\!48\)\( T^{7} + \)\(10\!\cdots\!81\)\( T^{8} \)
$71$ \( 1 + \)\(16\!\cdots\!32\)\( T + \)\(43\!\cdots\!08\)\( T^{2} + \)\(56\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!88\)\( T^{6} + \)\(92\!\cdots\!12\)\( T^{7} + \)\(99\!\cdots\!21\)\( T^{8} \)
$73$ \( 1 + \)\(51\!\cdots\!32\)\( T + \)\(33\!\cdots\!32\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!26\)\( T^{4} + \)\(24\!\cdots\!00\)\( T^{5} + \)\(85\!\cdots\!08\)\( T^{6} + \)\(20\!\cdots\!96\)\( T^{7} + \)\(64\!\cdots\!61\)\( T^{8} \)
$79$ \( 1 - \)\(37\!\cdots\!20\)\( T + \)\(83\!\cdots\!76\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(99\!\cdots\!60\)\( T^{5} + \)\(55\!\cdots\!36\)\( T^{6} - \)\(20\!\cdots\!80\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} \)
$83$ \( 1 - \)\(19\!\cdots\!08\)\( T + \)\(24\!\cdots\!12\)\( T^{2} - \)\(21\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!26\)\( T^{4} - \)\(88\!\cdots\!20\)\( T^{5} + \)\(39\!\cdots\!08\)\( T^{6} - \)\(12\!\cdots\!84\)\( T^{7} + \)\(26\!\cdots\!81\)\( T^{8} \)
$89$ \( 1 - \)\(45\!\cdots\!40\)\( T + \)\(11\!\cdots\!36\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!86\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!16\)\( T^{6} - \)\(46\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$97$ \( 1 - \)\(87\!\cdots\!64\)\( T + \)\(20\!\cdots\!68\)\( T^{2} - \)\(87\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!46\)\( T^{4} - \)\(78\!\cdots\!60\)\( T^{5} + \)\(16\!\cdots\!52\)\( T^{6} - \)\(64\!\cdots\!68\)\( T^{7} + \)\(66\!\cdots\!21\)\( T^{8} \)
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