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: \( 2^{14}\cdot 3^{6}\cdot 5 \)
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.

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

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

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

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.

Hecke kernels

There are no other newforms in \(S_{46}^{\mathrm{new}}(\Gamma_0(1))\).