Properties

Label 2.46.a.a
Level 2
Weight 46
Character orbit 2.a
Self dual Yes
Analytic conductor 25.651
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 5^{2}\cdot 11 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2851200\sqrt{655098313}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -4194304 q^{2} \) \( + ( -34883103276 - \beta ) q^{3} \) \( + 17592186044416 q^{4} \) \( + ( -2251248081340650 - 42476 \beta ) q^{5} \) \( + ( 146310339602939904 + 4194304 \beta ) q^{6} \) \( + ( -4782175712604552472 - 7308322 \beta ) q^{7} \) \( -73786976294838206464 q^{8} \) \( + ( 3588036050758238953533 + 69766206552 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(-4194304 q^{2}\) \(+(-34883103276 - \beta) q^{3}\) \(+17592186044416 q^{4}\) \(+(-2251248081340650 - 42476 \beta) q^{5}\) \(+(146310339602939904 + 4194304 \beta) q^{6}\) \(+(-4782175712604552472 - 7308322 \beta) q^{7}\) \(-73786976294838206464 q^{8}\) \(+(\)\(35\!\cdots\!33\)\( + 69766206552 \beta) q^{9}\) \(+(\)\(94\!\cdots\!00\)\( + 178157256704 \beta) q^{10}\) \(+(\)\(99\!\cdots\!32\)\( + 2762949838957 \beta) q^{11}\) \(+(-\)\(61\!\cdots\!16\)\( - 17592186044416 \beta) q^{12}\) \(+(\)\(57\!\cdots\!54\)\( - 127591092331532 \beta) q^{13}\) \(+(\)\(20\!\cdots\!88\)\( + 30653324197888 \beta) q^{14}\) \(+(\)\(30\!\cdots\!00\)\( + 3732942776092026 \beta) q^{15}\) \(+\)\(30\!\cdots\!56\)\( q^{16}\) \(+(\)\(78\!\cdots\!78\)\( - 7657971672330088 \beta) q^{17}\) \(+(-\)\(15\!\cdots\!32\)\( - 292620679205879808 \beta) q^{18}\) \(+(-\)\(19\!\cdots\!00\)\( + 943670096252337787 \beta) q^{19}\) \(+(-\)\(39\!\cdots\!00\)\( - 747245694422614016 \beta) q^{20}\) \(+(\)\(20\!\cdots\!72\)\( + 5037112663704815344 \beta) q^{21}\) \(+(-\)\(41\!\cdots\!28\)\( - 11588651561336700928 \beta) q^{22}\) \(+(-\)\(15\!\cdots\!36\)\( - 44126344403460381286 \beta) q^{23}\) \(+(\)\(25\!\cdots\!64\)\( + 73786976294838206464 \beta) q^{24}\) \(+(-\)\(13\!\cdots\!25\)\( + \)\(19\!\cdots\!00\)\( \beta) q^{25}\) \(+(-\)\(24\!\cdots\!16\)\( + \)\(53\!\cdots\!28\)\( \beta) q^{26}\) \(+(-\)\(39\!\cdots\!40\)\( - \)\(30\!\cdots\!42\)\( \beta) q^{27}\) \(+(-\)\(84\!\cdots\!52\)\( - \)\(12\!\cdots\!52\)\( \beta) q^{28}\) \(+(-\)\(14\!\cdots\!10\)\( + \)\(10\!\cdots\!44\)\( \beta) q^{29}\) \(+(-\)\(12\!\cdots\!00\)\( - \)\(15\!\cdots\!04\)\( \beta) q^{30}\) \(+(-\)\(20\!\cdots\!88\)\( + \)\(35\!\cdots\!24\)\( \beta) q^{31}\) \(-\)\(12\!\cdots\!24\)\( q^{32}\) \(+(-\)\(18\!\cdots\!32\)\( - \)\(19\!\cdots\!64\)\( \beta) q^{33}\) \(+(-\)\(32\!\cdots\!12\)\( + \)\(32\!\cdots\!52\)\( \beta) q^{34}\) \(+(\)\(12\!\cdots\!00\)\( + \)\(21\!\cdots\!72\)\( \beta) q^{35}\) \(+(\)\(63\!\cdots\!28\)\( + \)\(12\!\cdots\!32\)\( \beta) q^{36}\) \(+(\)\(25\!\cdots\!58\)\( - \)\(64\!\cdots\!28\)\( \beta) q^{37}\) \(+(\)\(79\!\cdots\!00\)\( - \)\(39\!\cdots\!48\)\( \beta) q^{38}\) \(+(\)\(47\!\cdots\!96\)\( - \)\(12\!\cdots\!22\)\( \beta) q^{39}\) \(+(\)\(16\!\cdots\!00\)\( + \)\(31\!\cdots\!64\)\( \beta) q^{40}\) \(+(\)\(20\!\cdots\!42\)\( + \)\(22\!\cdots\!56\)\( \beta) q^{41}\) \(+(-\)\(86\!\cdots\!88\)\( - \)\(21\!\cdots\!76\)\( \beta) q^{42}\) \(+(-\)\(17\!\cdots\!76\)\( + \)\(91\!\cdots\!37\)\( \beta) q^{43}\) \(+(\)\(17\!\cdots\!12\)\( + \)\(48\!\cdots\!12\)\( \beta) q^{44}\) \(+(-\)\(23\!\cdots\!50\)\( - \)\(30\!\cdots\!08\)\( \beta) q^{45}\) \(+(\)\(63\!\cdots\!44\)\( + \)\(18\!\cdots\!44\)\( \beta) q^{46}\) \(+(-\)\(60\!\cdots\!12\)\( + \)\(57\!\cdots\!88\)\( \beta) q^{47}\) \(+(-\)\(10\!\cdots\!56\)\( - \)\(30\!\cdots\!56\)\( \beta) q^{48}\) \(+(-\)\(83\!\cdots\!23\)\( + \)\(69\!\cdots\!68\)\( \beta) q^{49}\) \(+(\)\(57\!\cdots\!00\)\( - \)\(80\!\cdots\!00\)\( \beta) q^{50}\) \(+(\)\(13\!\cdots\!72\)\( - \)\(51\!\cdots\!90\)\( \beta) q^{51}\) \(+(\)\(10\!\cdots\!64\)\( - \)\(22\!\cdots\!12\)\( \beta) q^{52}\) \(+(\)\(70\!\cdots\!34\)\( + \)\(31\!\cdots\!88\)\( \beta) q^{53}\) \(+(\)\(16\!\cdots\!60\)\( + \)\(12\!\cdots\!68\)\( \beta) q^{54}\) \(+(-\)\(84\!\cdots\!00\)\( - \)\(10\!\cdots\!82\)\( \beta) q^{55}\) \(+(\)\(35\!\cdots\!08\)\( + \)\(53\!\cdots\!08\)\( \beta) q^{56}\) \(+(-\)\(43\!\cdots\!00\)\( - \)\(13\!\cdots\!12\)\( \beta) q^{57}\) \(+(\)\(58\!\cdots\!40\)\( - \)\(43\!\cdots\!76\)\( \beta) q^{58}\) \(+(-\)\(72\!\cdots\!20\)\( + \)\(30\!\cdots\!33\)\( \beta) q^{59}\) \(+(\)\(53\!\cdots\!00\)\( + \)\(65\!\cdots\!16\)\( \beta) q^{60}\) \(+(-\)\(10\!\cdots\!78\)\( + \)\(18\!\cdots\!28\)\( \beta) q^{61}\) \(+(\)\(87\!\cdots\!52\)\( - \)\(15\!\cdots\!96\)\( \beta) q^{62}\) \(+(-\)\(19\!\cdots\!76\)\( - \)\(35\!\cdots\!70\)\( \beta) q^{63}\) \(+\)\(54\!\cdots\!96\)\( q^{64}\) \(+(\)\(15\!\cdots\!00\)\( + \)\(44\!\cdots\!96\)\( \beta) q^{65}\) \(+(\)\(76\!\cdots\!28\)\( + \)\(82\!\cdots\!56\)\( \beta) q^{66}\) \(+(-\)\(48\!\cdots\!92\)\( - \)\(81\!\cdots\!81\)\( \beta) q^{67}\) \(+(\)\(13\!\cdots\!48\)\( - \)\(13\!\cdots\!08\)\( \beta) q^{68}\) \(+(\)\(28\!\cdots\!36\)\( + \)\(30\!\cdots\!72\)\( \beta) q^{69}\) \(+(-\)\(52\!\cdots\!00\)\( - \)\(92\!\cdots\!88\)\( \beta) q^{70}\) \(+(\)\(22\!\cdots\!72\)\( - \)\(43\!\cdots\!62\)\( \beta) q^{71}\) \(+(-\)\(26\!\cdots\!12\)\( - \)\(51\!\cdots\!28\)\( \beta) q^{72}\) \(+(-\)\(17\!\cdots\!26\)\( + \)\(56\!\cdots\!48\)\( \beta) q^{73}\) \(+(-\)\(10\!\cdots\!32\)\( + \)\(27\!\cdots\!12\)\( \beta) q^{74}\) \(+(-\)\(53\!\cdots\!00\)\( + \)\(70\!\cdots\!25\)\( \beta) q^{75}\) \(+(-\)\(33\!\cdots\!00\)\( + \)\(16\!\cdots\!92\)\( \beta) q^{76}\) \(+(-\)\(58\!\cdots\!04\)\( - \)\(13\!\cdots\!08\)\( \beta) q^{77}\) \(+(-\)\(20\!\cdots\!84\)\( + \)\(53\!\cdots\!88\)\( \beta) q^{78}\) \(+(-\)\(31\!\cdots\!40\)\( - \)\(12\!\cdots\!68\)\( \beta) q^{79}\) \(+(-\)\(69\!\cdots\!00\)\( - \)\(13\!\cdots\!56\)\( \beta) q^{80}\) \(+(\)\(19\!\cdots\!21\)\( + \)\(29\!\cdots\!96\)\( \beta) q^{81}\) \(+(-\)\(87\!\cdots\!68\)\( - \)\(96\!\cdots\!24\)\( \beta) q^{82}\) \(+(\)\(48\!\cdots\!24\)\( - \)\(11\!\cdots\!65\)\( \beta) q^{83}\) \(+(\)\(36\!\cdots\!52\)\( + \)\(88\!\cdots\!04\)\( \beta) q^{84}\) \(+(-\)\(27\!\cdots\!00\)\( - \)\(15\!\cdots\!28\)\( \beta) q^{85}\) \(+(\)\(74\!\cdots\!04\)\( - \)\(38\!\cdots\!48\)\( \beta) q^{86}\) \(+(-\)\(50\!\cdots\!40\)\( - \)\(22\!\cdots\!34\)\( \beta) q^{87}\) \(+(-\)\(73\!\cdots\!48\)\( - \)\(20\!\cdots\!48\)\( \beta) q^{88}\) \(+(-\)\(62\!\cdots\!10\)\( - \)\(26\!\cdots\!40\)\( \beta) q^{89}\) \(+(\)\(10\!\cdots\!00\)\( + \)\(12\!\cdots\!32\)\( \beta) q^{90}\) \(+(-\)\(22\!\cdots\!88\)\( + \)\(56\!\cdots\!16\)\( \beta) q^{91}\) \(+(-\)\(26\!\cdots\!76\)\( - \)\(77\!\cdots\!76\)\( \beta) q^{92}\) \(+(-\)\(11\!\cdots\!12\)\( + \)\(84\!\cdots\!64\)\( \beta) q^{93}\) \(+(\)\(25\!\cdots\!48\)\( - \)\(24\!\cdots\!52\)\( \beta) q^{94}\) \(+(-\)\(17\!\cdots\!00\)\( - \)\(13\!\cdots\!50\)\( \beta) q^{95}\) \(+(\)\(45\!\cdots\!24\)\( + \)\(12\!\cdots\!24\)\( \beta) q^{96}\) \(+(\)\(23\!\cdots\!18\)\( - \)\(26\!\cdots\!20\)\( \beta) q^{97}\) \(+(\)\(35\!\cdots\!92\)\( - \)\(29\!\cdots\!72\)\( \beta) q^{98}\) \(+(\)\(13\!\cdots\!56\)\( + \)\(16\!\cdots\!45\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 8388608q^{2} \) \(\mathstrut -\mathstrut 69766206552q^{3} \) \(\mathstrut +\mathstrut 35184372088832q^{4} \) \(\mathstrut -\mathstrut 4502496162681300q^{5} \) \(\mathstrut +\mathstrut 292620679205879808q^{6} \) \(\mathstrut -\mathstrut 9564351425209104944q^{7} \) \(\mathstrut -\mathstrut 147573952589676412928q^{8} \) \(\mathstrut +\mathstrut 7176072101516477907066q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 8388608q^{2} \) \(\mathstrut -\mathstrut 69766206552q^{3} \) \(\mathstrut +\mathstrut 35184372088832q^{4} \) \(\mathstrut -\mathstrut 4502496162681300q^{5} \) \(\mathstrut +\mathstrut 292620679205879808q^{6} \) \(\mathstrut -\mathstrut 9564351425209104944q^{7} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!28\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!66\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!64\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!32\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!08\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!76\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!12\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!56\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!64\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!44\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!56\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!72\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!28\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!32\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!80\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!04\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!20\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!76\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!48\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!64\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!24\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!56\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!16\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!92\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!84\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!76\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!52\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!24\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!00\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!88\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!24\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!12\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!46\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!44\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!28\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!68\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!20\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!16\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!00\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!40\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!56\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!04\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!52\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!92\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!56\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!84\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!96\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!72\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!44\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!24\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!52\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!64\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!08\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!68\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!80\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!42\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!36\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!48\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!04\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!08\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!96\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!76\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!52\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!24\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!96\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!48\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!36\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!84\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!12\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
12797.9
−12796.9
−4.19430e6 −1.07859e11 1.75922e13 −5.35098e15 4.52394e17 −5.31551e18 −7.37870e19 8.67930e21 2.24436e22
1.2 −4.19430e6 3.80930e10 1.75922e13 8.48487e14 −1.59774e17 −4.24884e18 −7.37870e19 −1.50323e21 −3.55881e21
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut +\mathstrut 69766206552 T_{3} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!24\)\( \) acting on \(S_{46}^{\mathrm{new}}(\Gamma_0(2))\).