Properties

Label 1.46.a.a
Level 1
Weight 46
Character orbit 1.a
Self dual yes
Analytic conductor 12.826
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 46 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.8255726074\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{14}\cdot 3^{6}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q + ( 1271424 + \beta_{1} ) q^{2} + ( 1786622292 - 923 \beta_{1} - \beta_{2} ) q^{3} + ( -638388297728 + 280176 \beta_{1} + 336 \beta_{2} ) q^{4} + ( -304149486153450 + 26067740 \beta_{1} + 113940 \beta_{2} ) q^{5} + ( -28134193689681408 - 31063479084 \beta_{1} - 3844224 \beta_{2} ) q^{6} + ( -2539903642212685336 - 1953973990038 \beta_{1} + 16353246 \beta_{2} ) q^{7} + ( -36315889252887429120 - 24755436464128 \beta_{1} + 1281595392 \beta_{2} ) q^{8} + ( 386233866896530598973 + 284518256889672 \beta_{1} - 32875127208 \beta_{2} ) q^{9} +O(q^{10})\) \( q +(1271424 + \beta_{1}) q^{2} +(1786622292 - 923 \beta_{1} - \beta_{2}) q^{3} +(-638388297728 + 280176 \beta_{1} + 336 \beta_{2}) q^{4} +(-304149486153450 + 26067740 \beta_{1} + 113940 \beta_{2}) q^{5} +(-28134193689681408 - 31063479084 \beta_{1} - 3844224 \beta_{2}) q^{6} +(-2539903642212685336 - 1953973990038 \beta_{1} + 16353246 \beta_{2}) q^{7} +(-36315889252887429120 - 24755436464128 \beta_{1} + 1281595392 \beta_{2}) q^{8} +(\)\(38\!\cdots\!73\)\( + 284518256889672 \beta_{1} - 32875127208 \beta_{2}) q^{9} +(\)\(47\!\cdots\!00\)\( + 3517197671230230 \beta_{1} + 411433658880 \beta_{2}) q^{10} +(-\)\(97\!\cdots\!48\)\( - 33130934920632065 \beta_{1} - 3038553063315 \beta_{2}) q^{11} +(-\)\(11\!\cdots\!16\)\( - 94667719590167360 \beta_{1} + 11161186455104 \beta_{2}) q^{12} +(-\)\(81\!\cdots\!18\)\( + 1425554182574644668 \beta_{1} + 23739268998900 \beta_{2}) q^{13} +(-\)\(67\!\cdots\!16\)\( - 50863100641931032 \beta_{1} - 598741319377152 \beta_{2}) q^{14} +(-\)\(37\!\cdots\!00\)\( - 29960911299403986210 \beta_{1} + 4049879334746490 \beta_{2}) q^{15} +(-\)\(83\!\cdots\!44\)\( + 21638169243366653952 \beta_{1} - 15610494525308928 \beta_{2}) q^{16} +(\)\(38\!\cdots\!94\)\( + \)\(53\!\cdots\!24\)\( \beta_{1} + 32920251664370328 \beta_{2}) q^{17} +(\)\(98\!\cdots\!92\)\( - \)\(10\!\cdots\!23\)\( \beta_{1} - 20585721250354176 \beta_{2}) q^{18} +(\)\(46\!\cdots\!60\)\( - \)\(48\!\cdots\!71\)\( \beta_{1} + 59078291628689019 \beta_{2}) q^{19} +(\)\(12\!\cdots\!00\)\( + \)\(99\!\cdots\!80\)\( \beta_{1} - 1373082890154988320 \beta_{2}) q^{20} +(\)\(75\!\cdots\!32\)\( + \)\(56\!\cdots\!52\)\( \beta_{1} + 8114319854144031472 \beta_{2}) q^{21} +(-\)\(12\!\cdots\!52\)\( - \)\(16\!\cdots\!28\)\( \beta_{1} - 21870532360181662080 \beta_{2}) q^{22} +(-\)\(35\!\cdots\!28\)\( - \)\(16\!\cdots\!46\)\( \beta_{1} + 8756374098281691546 \beta_{2}) q^{23} +(-\)\(35\!\cdots\!20\)\( + \)\(44\!\cdots\!72\)\( \beta_{1} + \)\(14\!\cdots\!92\)\( \beta_{2}) q^{24} +(\)\(14\!\cdots\!75\)\( + \)\(31\!\cdots\!00\)\( \beta_{1} - \)\(49\!\cdots\!00\)\( \beta_{2}) q^{25} +(\)\(36\!\cdots\!72\)\( - \)\(87\!\cdots\!82\)\( \beta_{1} + \)\(56\!\cdots\!48\)\( \beta_{2}) q^{26} +(\)\(95\!\cdots\!80\)\( + \)\(23\!\cdots\!22\)\( \beta_{1} + \)\(73\!\cdots\!42\)\( \beta_{2}) q^{27} +(\)\(17\!\cdots\!28\)\( - \)\(18\!\cdots\!24\)\( \beta_{1} - \)\(27\!\cdots\!16\)\( \beta_{2}) q^{28} +(-\)\(22\!\cdots\!10\)\( + \)\(14\!\cdots\!96\)\( \beta_{1} + \)\(46\!\cdots\!56\)\( \beta_{2}) q^{29} +(-\)\(14\!\cdots\!00\)\( - \)\(21\!\cdots\!20\)\( \beta_{1} + \)\(42\!\cdots\!80\)\( \beta_{2}) q^{30} +(-\)\(14\!\cdots\!48\)\( - \)\(13\!\cdots\!20\)\( \beta_{1} + \)\(19\!\cdots\!80\)\( \beta_{2}) q^{31} +(\)\(92\!\cdots\!24\)\( - \)\(51\!\cdots\!84\)\( \beta_{1} - \)\(92\!\cdots\!60\)\( \beta_{2}) q^{32} +(\)\(10\!\cdots\!84\)\( + \)\(18\!\cdots\!64\)\( \beta_{1} + \)\(81\!\cdots\!08\)\( \beta_{2}) q^{33} +(\)\(17\!\cdots\!44\)\( + \)\(97\!\cdots\!82\)\( \beta_{1} + \)\(29\!\cdots\!52\)\( \beta_{2}) q^{34} +(\)\(52\!\cdots\!00\)\( - \)\(64\!\cdots\!20\)\( \beta_{1} - \)\(87\!\cdots\!20\)\( \beta_{2}) q^{35} +(-\)\(34\!\cdots\!44\)\( + \)\(15\!\cdots\!12\)\( \beta_{1} + \)\(74\!\cdots\!32\)\( \beta_{2}) q^{36} +(-\)\(12\!\cdots\!46\)\( - \)\(43\!\cdots\!72\)\( \beta_{1} + \)\(33\!\cdots\!72\)\( \beta_{2}) q^{37} +(-\)\(99\!\cdots\!80\)\( + \)\(53\!\cdots\!88\)\( \beta_{1} - \)\(14\!\cdots\!32\)\( \beta_{2}) q^{38} +(-\)\(13\!\cdots\!44\)\( - \)\(41\!\cdots\!62\)\( \beta_{1} + \)\(52\!\cdots\!18\)\( \beta_{2}) q^{39} +(\)\(47\!\cdots\!00\)\( - \)\(52\!\cdots\!00\)\( \beta_{1} - \)\(15\!\cdots\!00\)\( \beta_{2}) q^{40} +(\)\(90\!\cdots\!02\)\( - \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2}) q^{41} +(\)\(18\!\cdots\!08\)\( + \)\(21\!\cdots\!96\)\( \beta_{1} + \)\(47\!\cdots\!84\)\( \beta_{2}) q^{42} +(\)\(41\!\cdots\!52\)\( + \)\(48\!\cdots\!51\)\( \beta_{1} - \)\(10\!\cdots\!87\)\( \beta_{2}) q^{43} +(-\)\(36\!\cdots\!56\)\( - \)\(62\!\cdots\!28\)\( \beta_{1} - \)\(26\!\cdots\!08\)\( \beta_{2}) q^{44} +(-\)\(12\!\cdots\!50\)\( + \)\(12\!\cdots\!20\)\( \beta_{1} + \)\(25\!\cdots\!20\)\( \beta_{2}) q^{45} +(-\)\(98\!\cdots\!48\)\( - \)\(31\!\cdots\!40\)\( \beta_{1} - \)\(22\!\cdots\!40\)\( \beta_{2}) q^{46} +(-\)\(17\!\cdots\!16\)\( + \)\(39\!\cdots\!20\)\( \beta_{1} - \)\(59\!\cdots\!16\)\( \beta_{2}) q^{47} +(\)\(49\!\cdots\!72\)\( + \)\(41\!\cdots\!04\)\( \beta_{1} + \)\(26\!\cdots\!96\)\( \beta_{2}) q^{48} +(\)\(26\!\cdots\!57\)\( - \)\(81\!\cdots\!20\)\( \beta_{1} + \)\(10\!\cdots\!80\)\( \beta_{2}) q^{49} +(\)\(12\!\cdots\!00\)\( - \)\(51\!\cdots\!25\)\( \beta_{1} - \)\(69\!\cdots\!00\)\( \beta_{2}) q^{50} +(-\)\(12\!\cdots\!88\)\( - \)\(24\!\cdots\!86\)\( \beta_{1} - \)\(65\!\cdots\!46\)\( \beta_{2}) q^{51} +(\)\(44\!\cdots\!64\)\( + \)\(14\!\cdots\!72\)\( \beta_{1} - \)\(17\!\cdots\!44\)\( \beta_{2}) q^{52} +(-\)\(61\!\cdots\!58\)\( + \)\(28\!\cdots\!12\)\( \beta_{1} + \)\(37\!\cdots\!84\)\( \beta_{2}) q^{53} +(\)\(19\!\cdots\!60\)\( + \)\(11\!\cdots\!84\)\( \beta_{1} + \)\(33\!\cdots\!24\)\( \beta_{2}) q^{54} +(-\)\(11\!\cdots\!00\)\( - \)\(19\!\cdots\!70\)\( \beta_{1} - \)\(75\!\cdots\!70\)\( \beta_{2}) q^{55} +(\)\(17\!\cdots\!60\)\( - \)\(69\!\cdots\!76\)\( \beta_{1} + \)\(51\!\cdots\!64\)\( \beta_{2}) q^{56} +(\)\(34\!\cdots\!60\)\( + \)\(85\!\cdots\!04\)\( \beta_{1} - \)\(32\!\cdots\!56\)\( \beta_{2}) q^{57} +(\)\(45\!\cdots\!80\)\( - \)\(35\!\cdots\!38\)\( \beta_{1} + \)\(50\!\cdots\!32\)\( \beta_{2}) q^{58} +(-\)\(11\!\cdots\!20\)\( + \)\(13\!\cdots\!67\)\( \beta_{1} + \)\(64\!\cdots\!37\)\( \beta_{2}) q^{59} +(\)\(44\!\cdots\!00\)\( - \)\(60\!\cdots\!20\)\( \beta_{1} - \)\(19\!\cdots\!20\)\( \beta_{2}) q^{60} +(-\)\(17\!\cdots\!98\)\( - \)\(17\!\cdots\!00\)\( \beta_{1} + \)\(78\!\cdots\!00\)\( \beta_{2}) q^{61} +(-\)\(18\!\cdots\!52\)\( - \)\(80\!\cdots\!88\)\( \beta_{1} + \)\(67\!\cdots\!60\)\( \beta_{2}) q^{62} +(-\)\(21\!\cdots\!88\)\( + \)\(19\!\cdots\!90\)\( \beta_{1} + \)\(77\!\cdots\!42\)\( \beta_{2}) q^{63} +(\)\(13\!\cdots\!12\)\( - \)\(24\!\cdots\!08\)\( \beta_{1} + \)\(47\!\cdots\!12\)\( \beta_{2}) q^{64} +(\)\(12\!\cdots\!00\)\( + \)\(53\!\cdots\!40\)\( \beta_{1} - \)\(64\!\cdots\!60\)\( \beta_{2}) q^{65} +(\)\(75\!\cdots\!84\)\( + \)\(11\!\cdots\!52\)\( \beta_{1} + \)\(91\!\cdots\!72\)\( \beta_{2}) q^{66} +(\)\(47\!\cdots\!44\)\( - \)\(30\!\cdots\!91\)\( \beta_{1} - \)\(10\!\cdots\!37\)\( \beta_{2}) q^{67} +(\)\(41\!\cdots\!88\)\( + \)\(82\!\cdots\!00\)\( \beta_{1} + \)\(21\!\cdots\!48\)\( \beta_{2}) q^{68} +(-\)\(30\!\cdots\!04\)\( + \)\(58\!\cdots\!24\)\( \beta_{1} + \)\(42\!\cdots\!64\)\( \beta_{2}) q^{69} +(-\)\(20\!\cdots\!00\)\( - \)\(17\!\cdots\!40\)\( \beta_{1} - \)\(52\!\cdots\!40\)\( \beta_{2}) q^{70} +(-\)\(41\!\cdots\!48\)\( + \)\(51\!\cdots\!50\)\( \beta_{1} - \)\(41\!\cdots\!50\)\( \beta_{2}) q^{71} +(-\)\(38\!\cdots\!60\)\( + \)\(26\!\cdots\!76\)\( \beta_{1} + \)\(34\!\cdots\!36\)\( \beta_{2}) q^{72} +(\)\(18\!\cdots\!22\)\( - \)\(44\!\cdots\!56\)\( \beta_{1} + \)\(14\!\cdots\!36\)\( \beta_{2}) q^{73} +(-\)\(30\!\cdots\!36\)\( - \)\(11\!\cdots\!30\)\( \beta_{1} - \)\(25\!\cdots\!80\)\( \beta_{2}) q^{74} +(\)\(15\!\cdots\!00\)\( + \)\(20\!\cdots\!75\)\( \beta_{1} - \)\(39\!\cdots\!75\)\( \beta_{2}) q^{75} +(-\)\(85\!\cdots\!80\)\( - \)\(30\!\cdots\!92\)\( \beta_{1} + \)\(10\!\cdots\!88\)\( \beta_{2}) q^{76} +(\)\(22\!\cdots\!28\)\( + \)\(29\!\cdots\!44\)\( \beta_{1} + \)\(41\!\cdots\!12\)\( \beta_{2}) q^{77} +(-\)\(15\!\cdots\!96\)\( + \)\(81\!\cdots\!72\)\( \beta_{1} + \)\(46\!\cdots\!96\)\( \beta_{2}) q^{78} +(\)\(30\!\cdots\!40\)\( + \)\(36\!\cdots\!36\)\( \beta_{1} - \)\(39\!\cdots\!04\)\( \beta_{2}) q^{79} +(-\)\(56\!\cdots\!00\)\( - \)\(36\!\cdots\!60\)\( \beta_{1} - \)\(25\!\cdots\!60\)\( \beta_{2}) q^{80} +(-\)\(34\!\cdots\!59\)\( - \)\(11\!\cdots\!24\)\( \beta_{1} + \)\(20\!\cdots\!36\)\( \beta_{2}) q^{81} +(-\)\(23\!\cdots\!52\)\( + \)\(14\!\cdots\!62\)\( \beta_{1} + \)\(67\!\cdots\!60\)\( \beta_{2}) q^{82} +(\)\(49\!\cdots\!12\)\( - \)\(54\!\cdots\!95\)\( \beta_{1} + \)\(73\!\cdots\!07\)\( \beta_{2}) q^{83} +(\)\(95\!\cdots\!04\)\( + \)\(12\!\cdots\!56\)\( \beta_{1} - \)\(43\!\cdots\!84\)\( \beta_{2}) q^{84} +(\)\(12\!\cdots\!00\)\( + \)\(27\!\cdots\!80\)\( \beta_{1} + \)\(50\!\cdots\!80\)\( \beta_{2}) q^{85} +(\)\(16\!\cdots\!12\)\( - \)\(35\!\cdots\!56\)\( \beta_{1} - \)\(20\!\cdots\!16\)\( \beta_{2}) q^{86} +(-\)\(63\!\cdots\!60\)\( - \)\(42\!\cdots\!54\)\( \beta_{1} - \)\(13\!\cdots\!94\)\( \beta_{2}) q^{87} +(\)\(17\!\cdots\!60\)\( + \)\(19\!\cdots\!44\)\( \beta_{1} + \)\(46\!\cdots\!84\)\( \beta_{2}) q^{88} +(-\)\(52\!\cdots\!30\)\( + \)\(10\!\cdots\!28\)\( \beta_{1} - \)\(21\!\cdots\!92\)\( \beta_{2}) q^{89} +(-\)\(11\!\cdots\!00\)\( - \)\(37\!\cdots\!10\)\( \beta_{1} + \)\(93\!\cdots\!40\)\( \beta_{2}) q^{90} +(-\)\(69\!\cdots\!88\)\( + \)\(17\!\cdots\!24\)\( \beta_{1} - \)\(11\!\cdots\!36\)\( \beta_{2}) q^{91} +(\)\(10\!\cdots\!44\)\( - \)\(18\!\cdots\!56\)\( \beta_{1} - \)\(14\!\cdots\!52\)\( \beta_{2}) q^{92} +(-\)\(66\!\cdots\!16\)\( - \)\(35\!\cdots\!16\)\( \beta_{1} + \)\(21\!\cdots\!28\)\( \beta_{2}) q^{93} +(\)\(12\!\cdots\!24\)\( - \)\(25\!\cdots\!56\)\( \beta_{1} - \)\(79\!\cdots\!16\)\( \beta_{2}) q^{94} +(\)\(39\!\cdots\!00\)\( - \)\(13\!\cdots\!50\)\( \beta_{1} + \)\(37\!\cdots\!50\)\( \beta_{2}) q^{95} +(\)\(32\!\cdots\!92\)\( + \)\(38\!\cdots\!56\)\( \beta_{1} - \)\(27\!\cdots\!84\)\( \beta_{2}) q^{96} +(\)\(12\!\cdots\!34\)\( - \)\(25\!\cdots\!40\)\( \beta_{1} - \)\(47\!\cdots\!76\)\( \beta_{2}) q^{97} +(\)\(64\!\cdots\!68\)\( + \)\(62\!\cdots\!17\)\( \beta_{1} + \)\(34\!\cdots\!60\)\( \beta_{2}) q^{98} +(-\)\(17\!\cdots\!04\)\( + \)\(11\!\cdots\!99\)\( \beta_{1} - \)\(39\!\cdots\!11\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3814272q^{2} + 5359866876q^{3} - 1915164893184q^{4} - 912448458460350q^{5} - 84402581069044224q^{6} - 7619710926638056008q^{7} - 108947667758662287360q^{8} + 1158701600689591796919q^{9} + O(q^{10}) \) \( 3q + 3814272q^{2} + 5359866876q^{3} - 1915164893184q^{4} - 912448458460350q^{5} - 84402581069044224q^{6} - 7619710926638056008q^{7} - \)\(10\!\cdots\!60\)\(q^{8} + \)\(11\!\cdots\!19\)\(q^{9} + \)\(14\!\cdots\!00\)\(q^{10} - \)\(29\!\cdots\!44\)\(q^{11} - \)\(33\!\cdots\!48\)\(q^{12} - \)\(24\!\cdots\!54\)\(q^{13} - \)\(20\!\cdots\!48\)\(q^{14} - \)\(11\!\cdots\!00\)\(q^{15} - \)\(25\!\cdots\!32\)\(q^{16} + \)\(11\!\cdots\!82\)\(q^{17} + \)\(29\!\cdots\!76\)\(q^{18} + \)\(13\!\cdots\!80\)\(q^{19} + \)\(38\!\cdots\!00\)\(q^{20} + \)\(22\!\cdots\!96\)\(q^{21} - \)\(36\!\cdots\!56\)\(q^{22} - \)\(10\!\cdots\!84\)\(q^{23} - \)\(10\!\cdots\!60\)\(q^{24} + \)\(43\!\cdots\!25\)\(q^{25} + \)\(10\!\cdots\!16\)\(q^{26} + \)\(28\!\cdots\!40\)\(q^{27} + \)\(53\!\cdots\!84\)\(q^{28} - \)\(67\!\cdots\!30\)\(q^{29} - \)\(44\!\cdots\!00\)\(q^{30} - \)\(43\!\cdots\!44\)\(q^{31} + \)\(27\!\cdots\!72\)\(q^{32} + \)\(32\!\cdots\!52\)\(q^{33} + \)\(53\!\cdots\!32\)\(q^{34} + \)\(15\!\cdots\!00\)\(q^{35} - \)\(10\!\cdots\!32\)\(q^{36} - \)\(38\!\cdots\!38\)\(q^{37} - \)\(29\!\cdots\!40\)\(q^{38} - \)\(40\!\cdots\!32\)\(q^{39} + \)\(14\!\cdots\!00\)\(q^{40} + \)\(27\!\cdots\!06\)\(q^{41} + \)\(55\!\cdots\!24\)\(q^{42} + \)\(12\!\cdots\!56\)\(q^{43} - \)\(10\!\cdots\!68\)\(q^{44} - \)\(36\!\cdots\!50\)\(q^{45} - \)\(29\!\cdots\!44\)\(q^{46} - \)\(53\!\cdots\!48\)\(q^{47} + \)\(14\!\cdots\!16\)\(q^{48} + \)\(78\!\cdots\!71\)\(q^{49} + \)\(36\!\cdots\!00\)\(q^{50} - \)\(37\!\cdots\!64\)\(q^{51} + \)\(13\!\cdots\!92\)\(q^{52} - \)\(18\!\cdots\!74\)\(q^{53} + \)\(59\!\cdots\!80\)\(q^{54} - \)\(34\!\cdots\!00\)\(q^{55} + \)\(52\!\cdots\!80\)\(q^{56} + \)\(10\!\cdots\!80\)\(q^{57} + \)\(13\!\cdots\!40\)\(q^{58} - \)\(35\!\cdots\!60\)\(q^{59} + \)\(13\!\cdots\!00\)\(q^{60} - \)\(51\!\cdots\!94\)\(q^{61} - \)\(56\!\cdots\!56\)\(q^{62} - \)\(63\!\cdots\!64\)\(q^{63} + \)\(41\!\cdots\!36\)\(q^{64} + \)\(37\!\cdots\!00\)\(q^{65} + \)\(22\!\cdots\!52\)\(q^{66} + \)\(14\!\cdots\!32\)\(q^{67} + \)\(12\!\cdots\!64\)\(q^{68} - \)\(91\!\cdots\!12\)\(q^{69} - \)\(61\!\cdots\!00\)\(q^{70} - \)\(12\!\cdots\!44\)\(q^{71} - \)\(11\!\cdots\!80\)\(q^{72} + \)\(54\!\cdots\!66\)\(q^{73} - \)\(92\!\cdots\!08\)\(q^{74} + \)\(46\!\cdots\!00\)\(q^{75} - \)\(25\!\cdots\!40\)\(q^{76} + \)\(66\!\cdots\!84\)\(q^{77} - \)\(45\!\cdots\!88\)\(q^{78} + \)\(90\!\cdots\!20\)\(q^{79} - \)\(16\!\cdots\!00\)\(q^{80} - \)\(10\!\cdots\!77\)\(q^{81} - \)\(70\!\cdots\!56\)\(q^{82} + \)\(14\!\cdots\!36\)\(q^{83} + \)\(28\!\cdots\!12\)\(q^{84} + \)\(38\!\cdots\!00\)\(q^{85} + \)\(49\!\cdots\!36\)\(q^{86} - \)\(19\!\cdots\!80\)\(q^{87} + \)\(52\!\cdots\!80\)\(q^{88} - \)\(15\!\cdots\!90\)\(q^{89} - \)\(34\!\cdots\!00\)\(q^{90} - \)\(20\!\cdots\!64\)\(q^{91} + \)\(31\!\cdots\!32\)\(q^{92} - \)\(19\!\cdots\!48\)\(q^{93} + \)\(38\!\cdots\!72\)\(q^{94} + \)\(11\!\cdots\!00\)\(q^{95} + \)\(97\!\cdots\!76\)\(q^{96} + \)\(37\!\cdots\!02\)\(q^{97} + \)\(19\!\cdots\!04\)\(q^{98} - \)\(52\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 148878150 x + 389915850150\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 576 \nu - 192 \)
\(\beta_{2}\)\(=\)\((\)\( 6912 \nu^{2} + 27147456 \nu - 686039566656 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 192\)\()/576\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{2} - 47131 \beta_{1} + 686030517504\)\()/6912\)

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
−13344.8
2760.23
10585.6
−6.41537e6 −1.72036e10 5.97255e12 2.46761e15 1.10367e17 1.29065e19 1.87405e20 −2.65835e21 −1.58306e22
1.2 2.86113e6 8.00971e10 −2.69983e13 −9.35259e15 2.29168e17 −6.95076e18 −1.77913e20 3.46124e21 −2.67589e22
1.3 7.36851e6 −5.75337e10 1.91106e13 5.97253e15 −4.23938e17 −1.35754e19 −1.18440e20 3.55813e20 4.40087e22
\(n\): 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 1.46.a.a 3
3.b odd 2 1 9.46.a.b 3
4.b odd 2 1 16.46.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.46.a.a 3 1.a even 1 1 trivial
9.46.a.b 3 3.b odd 2 1
16.46.a.c 3 4.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{46}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3814272 T + 61008476024832 T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!24\)\( T^{4} - \)\(47\!\cdots\!28\)\( T^{5} + \)\(43\!\cdots\!68\)\( T^{6} \)
$3$ \( 1 - 5359866876 T + \)\(38\!\cdots\!93\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!99\)\( T^{4} - \)\(46\!\cdots\!24\)\( T^{5} + \)\(25\!\cdots\!07\)\( T^{6} \)
$5$ \( 1 + 912448458460350 T + \)\(21\!\cdots\!75\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!75\)\( T^{4} + \)\(73\!\cdots\!50\)\( T^{5} + \)\(22\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 + 7619710926638056008 T + \)\(15\!\cdots\!57\)\( T^{2} + \)\(41\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!99\)\( T^{4} + \)\(87\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + \)\(29\!\cdots\!44\)\( T + \)\(14\!\cdots\!65\)\( T^{2} + \)\(45\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} + \)\(15\!\cdots\!44\)\( T^{5} + \)\(38\!\cdots\!51\)\( T^{6} \)
$13$ \( 1 + \)\(24\!\cdots\!54\)\( T + \)\(49\!\cdots\!43\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} + \)\(66\!\cdots\!99\)\( T^{4} + \)\(43\!\cdots\!46\)\( T^{5} + \)\(24\!\cdots\!57\)\( T^{6} \)
$17$ \( 1 - \)\(11\!\cdots\!82\)\( T + \)\(51\!\cdots\!07\)\( T^{2} - \)\(76\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!99\)\( T^{4} - \)\(64\!\cdots\!18\)\( T^{5} + \)\(12\!\cdots\!93\)\( T^{6} \)
$19$ \( 1 - \)\(13\!\cdots\!80\)\( T + \)\(15\!\cdots\!97\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(55\!\cdots\!03\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(42\!\cdots\!99\)\( T^{6} \)
$23$ \( 1 + \)\(10\!\cdots\!84\)\( T + \)\(93\!\cdots\!93\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!99\)\( T^{4} + \)\(38\!\cdots\!16\)\( T^{5} + \)\(68\!\cdots\!07\)\( T^{6} \)
$29$ \( 1 + \)\(67\!\cdots\!30\)\( T + \)\(10\!\cdots\!47\)\( T^{2} + \)\(83\!\cdots\!40\)\( T^{3} + \)\(65\!\cdots\!03\)\( T^{4} + \)\(27\!\cdots\!30\)\( T^{5} + \)\(26\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 + \)\(43\!\cdots\!44\)\( T + \)\(43\!\cdots\!65\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!15\)\( T^{4} + \)\(72\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 + \)\(38\!\cdots\!38\)\( T + \)\(16\!\cdots\!07\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!99\)\( T^{4} + \)\(53\!\cdots\!62\)\( T^{5} + \)\(50\!\cdots\!93\)\( T^{6} \)
$41$ \( 1 - \)\(27\!\cdots\!06\)\( T + \)\(12\!\cdots\!15\)\( T^{2} - \)\(20\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!15\)\( T^{4} - \)\(38\!\cdots\!06\)\( T^{5} + \)\(53\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 - \)\(12\!\cdots\!56\)\( T + \)\(31\!\cdots\!93\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(32\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 + \)\(53\!\cdots\!48\)\( T + \)\(27\!\cdots\!57\)\( T^{2} - \)\(60\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!99\)\( T^{4} + \)\(16\!\cdots\!52\)\( T^{5} + \)\(54\!\cdots\!43\)\( T^{6} \)
$53$ \( 1 + \)\(18\!\cdots\!74\)\( T + \)\(22\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!99\)\( T^{4} + \)\(28\!\cdots\!26\)\( T^{5} + \)\(59\!\cdots\!57\)\( T^{6} \)
$59$ \( 1 + \)\(35\!\cdots\!60\)\( T + \)\(46\!\cdots\!97\)\( T^{2} - \)\(62\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!03\)\( T^{4} + \)\(85\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 + \)\(51\!\cdots\!94\)\( T + \)\(13\!\cdots\!15\)\( T^{2} + \)\(23\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!15\)\( T^{4} + \)\(24\!\cdots\!94\)\( T^{5} + \)\(10\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 - \)\(14\!\cdots\!32\)\( T - \)\(71\!\cdots\!43\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} - \)\(10\!\cdots\!01\)\( T^{4} - \)\(31\!\cdots\!68\)\( T^{5} + \)\(33\!\cdots\!43\)\( T^{6} \)
$71$ \( 1 + \)\(12\!\cdots\!44\)\( T + \)\(39\!\cdots\!65\)\( T^{2} + \)\(60\!\cdots\!80\)\( T^{3} + \)\(80\!\cdots\!15\)\( T^{4} + \)\(51\!\cdots\!44\)\( T^{5} + \)\(83\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 - \)\(54\!\cdots\!66\)\( T + \)\(10\!\cdots\!43\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!99\)\( T^{4} - \)\(27\!\cdots\!34\)\( T^{5} + \)\(35\!\cdots\!57\)\( T^{6} \)
$79$ \( 1 - \)\(90\!\cdots\!20\)\( T + \)\(66\!\cdots\!97\)\( T^{2} - \)\(49\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!03\)\( T^{4} - \)\(55\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 - \)\(14\!\cdots\!36\)\( T + \)\(64\!\cdots\!93\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!99\)\( T^{4} - \)\(77\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!07\)\( T^{6} \)
$89$ \( 1 + \)\(15\!\cdots\!90\)\( T + \)\(23\!\cdots\!47\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!03\)\( T^{4} + \)\(43\!\cdots\!90\)\( T^{5} + \)\(14\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 - \)\(37\!\cdots\!02\)\( T + \)\(66\!\cdots\!07\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!99\)\( T^{4} - \)\(23\!\cdots\!98\)\( T^{5} + \)\(16\!\cdots\!93\)\( T^{6} \)
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