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: \( 2^{45}\cdot 3^{12}\cdot 5^{5}\cdot 7\cdot 11\cdot 13 \)
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.

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

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

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

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.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!88\)\( T_{3}^{3} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!96\)\( T_{3}^{2} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!08\)\( T_{3} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!44\)\( \) acting on \(S_{80}^{\mathrm{new}}(\Gamma_0(2))\).