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: \( 2^{23}\cdot 3^{9}\cdot 5^{4}\cdot 7\cdot 11 \)
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.

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

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

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

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.

Atkin-Lehner signs

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

Hecke kernels

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