Properties

Label 2.46.a.b
Level 2
Weight 46
Character orbit 2.a
Self dual Yes
Analytic conductor 25.651
Analytic rank 0
Dimension 2
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 46 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(25.6511452149\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 1920\sqrt{800679089088649}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 4194304 q^{2} \) \( + ( 29930608596 - \beta ) q^{3} \) \( + 17592186044416 q^{4} \) \( + ( 2198556995702550 - 127980 \beta ) q^{5} \) \( + ( 125538071356637184 - 4194304 \beta ) q^{6} \) \( + ( 3951246596735649512 + 28459998 \beta ) q^{7} \) \( + 73786976294838206464 q^{8} \) \( + ( 893152018392511066173 - 59861217192 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+4194304 q^{2}\) \(+(29930608596 - \beta) q^{3}\) \(+17592186044416 q^{4}\) \(+(2198556995702550 - 127980 \beta) q^{5}\) \(+(125538071356637184 - 4194304 \beta) q^{6}\) \(+(3951246596735649512 + 28459998 \beta) q^{7}\) \(+73786976294838206464 q^{8}\) \(+(\)\(89\!\cdots\!73\)\( - 59861217192 \beta) q^{9}\) \(+(\)\(92\!\cdots\!00\)\( - 536787025920 \beta) q^{10}\) \(+(\)\(96\!\cdots\!72\)\( + 2197121767533 \beta) q^{11}\) \(+(\)\(52\!\cdots\!36\)\( - 17592186044416 \beta) q^{12}\) \(+(-\)\(11\!\cdots\!14\)\( + 239553473871348 \beta) q^{13}\) \(+(\)\(16\!\cdots\!48\)\( + 119369883451392 \beta) q^{14}\) \(+(\)\(44\!\cdots\!00\)\( - 6029076283818630 \beta) q^{15}\) \(+\)\(30\!\cdots\!56\)\( q^{16}\) \(+(-\)\(32\!\cdots\!18\)\( + 91984363822443672 \beta) q^{17}\) \(+(\)\(37\!\cdots\!92\)\( - 251076142713274368 \beta) q^{18}\) \(+(-\)\(45\!\cdots\!60\)\( + 815181673524211323 \beta) q^{19}\) \(+(\)\(38\!\cdots\!00\)\( - 2251447969964359680 \beta) q^{20}\) \(+(\)\(34\!\cdots\!52\)\( - 3099421535954706704 \beta) q^{21}\) \(+(\)\(40\!\cdots\!88\)\( + 9215396618050732032 \beta) q^{22}\) \(+(\)\(34\!\cdots\!56\)\( + 63330158869140020634 \beta) q^{23}\) \(+(\)\(22\!\cdots\!44\)\( - 73786976294838206464 \beta) q^{24}\) \(+(\)\(24\!\cdots\!75\)\( - \)\(56\!\cdots\!00\)\( \beta) q^{25}\) \(+(-\)\(48\!\cdots\!56\)\( + \)\(10\!\cdots\!92\)\( \beta) q^{26}\) \(+(\)\(11\!\cdots\!80\)\( + \)\(26\!\cdots\!38\)\( \beta) q^{27}\) \(+(\)\(69\!\cdots\!92\)\( + \)\(50\!\cdots\!68\)\( \beta) q^{28}\) \(+(\)\(11\!\cdots\!70\)\( + \)\(23\!\cdots\!16\)\( \beta) q^{29}\) \(+(\)\(18\!\cdots\!00\)\( - \)\(25\!\cdots\!20\)\( \beta) q^{30}\) \(+(\)\(48\!\cdots\!92\)\( + \)\(28\!\cdots\!16\)\( \beta) q^{31}\) \(+\)\(12\!\cdots\!24\)\( q^{32}\) \(+(-\)\(35\!\cdots\!88\)\( - \)\(30\!\cdots\!04\)\( \beta) q^{33}\) \(+(-\)\(13\!\cdots\!72\)\( + \)\(38\!\cdots\!88\)\( \beta) q^{34}\) \(+(-\)\(20\!\cdots\!00\)\( - \)\(44\!\cdots\!60\)\( \beta) q^{35}\) \(+(\)\(15\!\cdots\!68\)\( - \)\(10\!\cdots\!72\)\( \beta) q^{36}\) \(+(-\)\(95\!\cdots\!98\)\( - \)\(33\!\cdots\!68\)\( \beta) q^{37}\) \(+(-\)\(18\!\cdots\!40\)\( + \)\(34\!\cdots\!92\)\( \beta) q^{38}\) \(+(-\)\(74\!\cdots\!44\)\( + \)\(83\!\cdots\!22\)\( \beta) q^{39}\) \(+(\)\(16\!\cdots\!00\)\( - \)\(94\!\cdots\!20\)\( \beta) q^{40}\) \(+(-\)\(92\!\cdots\!38\)\( - \)\(26\!\cdots\!76\)\( \beta) q^{41}\) \(+(\)\(14\!\cdots\!08\)\( - \)\(12\!\cdots\!16\)\( \beta) q^{42}\) \(+(-\)\(40\!\cdots\!64\)\( + \)\(98\!\cdots\!57\)\( \beta) q^{43}\) \(+(\)\(17\!\cdots\!52\)\( + \)\(38\!\cdots\!28\)\( \beta) q^{44}\) \(+(\)\(24\!\cdots\!50\)\( - \)\(24\!\cdots\!40\)\( \beta) q^{45}\) \(+(\)\(14\!\cdots\!24\)\( + \)\(26\!\cdots\!36\)\( \beta) q^{46}\) \(+(\)\(18\!\cdots\!52\)\( - \)\(84\!\cdots\!72\)\( \beta) q^{47}\) \(+(\)\(92\!\cdots\!76\)\( - \)\(30\!\cdots\!56\)\( \beta) q^{48}\) \(+(-\)\(89\!\cdots\!63\)\( + \)\(22\!\cdots\!52\)\( \beta) q^{49}\) \(+(\)\(10\!\cdots\!00\)\( - \)\(23\!\cdots\!00\)\( \beta) q^{50}\) \(+(-\)\(36\!\cdots\!28\)\( + \)\(60\!\cdots\!30\)\( \beta) q^{51}\) \(+(-\)\(20\!\cdots\!24\)\( + \)\(42\!\cdots\!68\)\( \beta) q^{52}\) \(+(-\)\(33\!\cdots\!94\)\( - \)\(15\!\cdots\!52\)\( \beta) q^{53}\) \(+(\)\(48\!\cdots\!20\)\( + \)\(11\!\cdots\!52\)\( \beta) q^{54}\) \(+(-\)\(61\!\cdots\!00\)\( - \)\(75\!\cdots\!10\)\( \beta) q^{55}\) \(+(\)\(29\!\cdots\!68\)\( + \)\(20\!\cdots\!72\)\( \beta) q^{56}\) \(+(-\)\(37\!\cdots\!60\)\( + \)\(69\!\cdots\!68\)\( \beta) q^{57}\) \(+(\)\(46\!\cdots\!80\)\( + \)\(97\!\cdots\!64\)\( \beta) q^{58}\) \(+(-\)\(73\!\cdots\!60\)\( - \)\(12\!\cdots\!63\)\( \beta) q^{59}\) \(+(\)\(78\!\cdots\!00\)\( - \)\(10\!\cdots\!80\)\( \beta) q^{60}\) \(+(\)\(84\!\cdots\!82\)\( + \)\(17\!\cdots\!52\)\( \beta) q^{61}\) \(+(\)\(20\!\cdots\!68\)\( + \)\(11\!\cdots\!64\)\( \beta) q^{62}\) \(+(-\)\(14\!\cdots\!24\)\( - \)\(21\!\cdots\!50\)\( \beta) q^{63}\) \(+\)\(54\!\cdots\!96\)\( q^{64}\) \(+(-\)\(93\!\cdots\!00\)\( + \)\(67\!\cdots\!20\)\( \beta) q^{65}\) \(+(-\)\(15\!\cdots\!52\)\( - \)\(12\!\cdots\!16\)\( \beta) q^{66}\) \(+(-\)\(41\!\cdots\!88\)\( - \)\(17\!\cdots\!41\)\( \beta) q^{67}\) \(+(-\)\(57\!\cdots\!88\)\( + \)\(16\!\cdots\!52\)\( \beta) q^{68}\) \(+(-\)\(82\!\cdots\!24\)\( - \)\(16\!\cdots\!92\)\( \beta) q^{69}\) \(+(-\)\(86\!\cdots\!00\)\( - \)\(18\!\cdots\!40\)\( \beta) q^{70}\) \(+(\)\(55\!\cdots\!32\)\( + \)\(77\!\cdots\!02\)\( \beta) q^{71}\) \(+(\)\(65\!\cdots\!72\)\( - \)\(44\!\cdots\!88\)\( \beta) q^{72}\) \(+(\)\(68\!\cdots\!66\)\( + \)\(10\!\cdots\!08\)\( \beta) q^{73}\) \(+(-\)\(40\!\cdots\!92\)\( - \)\(13\!\cdots\!72\)\( \beta) q^{74}\) \(+(\)\(24\!\cdots\!00\)\( - \)\(41\!\cdots\!75\)\( \beta) q^{75}\) \(+(-\)\(79\!\cdots\!60\)\( + \)\(14\!\cdots\!68\)\( \beta) q^{76}\) \(+(\)\(56\!\cdots\!64\)\( + \)\(11\!\cdots\!52\)\( \beta) q^{77}\) \(+(-\)\(31\!\cdots\!76\)\( + \)\(34\!\cdots\!88\)\( \beta) q^{78}\) \(+(\)\(16\!\cdots\!00\)\( + \)\(19\!\cdots\!88\)\( \beta) q^{79}\) \(+(\)\(68\!\cdots\!00\)\( - \)\(39\!\cdots\!80\)\( \beta) q^{80}\) \(+(\)\(78\!\cdots\!41\)\( + \)\(69\!\cdots\!24\)\( \beta) q^{81}\) \(+(-\)\(38\!\cdots\!52\)\( - \)\(11\!\cdots\!04\)\( \beta) q^{82}\) \(+(-\)\(47\!\cdots\!24\)\( - \)\(22\!\cdots\!05\)\( \beta) q^{83}\) \(+(\)\(60\!\cdots\!32\)\( - \)\(54\!\cdots\!64\)\( \beta) q^{84}\) \(+(-\)\(41\!\cdots\!00\)\( + \)\(62\!\cdots\!40\)\( \beta) q^{85}\) \(+(-\)\(16\!\cdots\!56\)\( + \)\(41\!\cdots\!28\)\( \beta) q^{86}\) \(+(\)\(26\!\cdots\!20\)\( - \)\(10\!\cdots\!34\)\( \beta) q^{87}\) \(+(\)\(71\!\cdots\!08\)\( + \)\(16\!\cdots\!12\)\( \beta) q^{88}\) \(+(\)\(33\!\cdots\!90\)\( + \)\(45\!\cdots\!40\)\( \beta) q^{89}\) \(+(\)\(10\!\cdots\!00\)\( - \)\(10\!\cdots\!60\)\( \beta) q^{90}\) \(+(\)\(15\!\cdots\!32\)\( + \)\(91\!\cdots\!04\)\( \beta) q^{91}\) \(+(\)\(61\!\cdots\!96\)\( + \)\(11\!\cdots\!44\)\( \beta) q^{92}\) \(+(\)\(61\!\cdots\!32\)\( - \)\(39\!\cdots\!56\)\( \beta) q^{93}\) \(+(\)\(75\!\cdots\!08\)\( - \)\(35\!\cdots\!88\)\( \beta) q^{94}\) \(+(-\)\(40\!\cdots\!00\)\( + \)\(75\!\cdots\!50\)\( \beta) q^{95}\) \(+(\)\(38\!\cdots\!04\)\( - \)\(12\!\cdots\!24\)\( \beta) q^{96}\) \(+(-\)\(41\!\cdots\!18\)\( - \)\(11\!\cdots\!80\)\( \beta) q^{97}\) \(+(-\)\(37\!\cdots\!52\)\( + \)\(94\!\cdots\!08\)\( \beta) q^{98}\) \(+(-\)\(30\!\cdots\!44\)\( - \)\(38\!\cdots\!15\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 8388608q^{2} \) \(\mathstrut +\mathstrut 59861217192q^{3} \) \(\mathstrut +\mathstrut 35184372088832q^{4} \) \(\mathstrut +\mathstrut 4397113991405100q^{5} \) \(\mathstrut +\mathstrut 251076142713274368q^{6} \) \(\mathstrut +\mathstrut 7902493193471299024q^{7} \) \(\mathstrut +\mathstrut 147573952589676412928q^{8} \) \(\mathstrut +\mathstrut 1786304036785022132346q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 8388608q^{2} \) \(\mathstrut +\mathstrut 59861217192q^{3} \) \(\mathstrut +\mathstrut 35184372088832q^{4} \) \(\mathstrut +\mathstrut 4397113991405100q^{5} \) \(\mathstrut +\mathstrut 251076142713274368q^{6} \) \(\mathstrut +\mathstrut 7902493193471299024q^{7} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!28\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!46\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!44\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!72\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!28\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!96\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!12\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!36\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!84\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!20\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!04\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!76\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!12\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!88\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!12\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!60\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!84\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!40\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!84\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!48\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!76\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!44\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!36\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!96\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!80\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!88\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!76\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!16\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!28\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!04\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!00\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!48\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!04\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!52\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!26\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!56\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!48\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!88\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!40\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!36\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!60\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!20\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!64\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!36\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!48\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!92\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!04\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!76\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!76\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!48\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!64\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!44\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!32\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!84\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!20\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!28\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!52\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!82\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!04\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!48\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!64\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!12\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!40\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!16\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!80\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!64\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!92\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!64\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!16\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!08\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!36\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!04\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.41481e7
−1.41481e7
4.19430e6 −2.43982e10 1.75922e13 −4.75445e15 −1.02334e17 5.49745e18 7.37870e19 −2.35904e21 −1.99416e22
1.2 4.19430e6 8.42595e10 1.75922e13 9.15156e15 3.53410e17 2.40505e18 7.37870e19 4.14534e21 3.83844e22
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 59861217192 T_{3} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!84\)\( \) acting on \(S_{46}^{\mathrm{new}}(\Gamma_0(2))\).