Properties

Label 2.66.a.a
Level 2
Weight 66
Character orbit 2.a
Self dual Yes
Analytic conductor 53.514
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 66 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(53.5144712945\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4}\cdot 5\cdot 11 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 285120\sqrt{7845027215820649}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4294967296 q^{2} +(574529980102596 - 111 \beta) q^{3} +18446744073709551616 q^{4} +(-\)\(37\!\cdots\!50\)\( + 2449901020 \beta) q^{5} +(-\)\(24\!\cdots\!16\)\( + 476741369856 \beta) q^{6} +(\)\(13\!\cdots\!12\)\( - 18515264307022 \beta) q^{7} -\)\(79\!\cdots\!36\)\( q^{8} +(-\)\(21\!\cdots\!27\)\( - 127545655582776312 \beta) q^{9} +O(q^{10})\) \( q -4294967296 q^{2} +(574529980102596 - 111 \beta) q^{3} +18446744073709551616 q^{4} +(-\)\(37\!\cdots\!50\)\( + 2449901020 \beta) q^{5} +(-\)\(24\!\cdots\!16\)\( + 476741369856 \beta) q^{6} +(\)\(13\!\cdots\!12\)\( - 18515264307022 \beta) q^{7} -\)\(79\!\cdots\!36\)\( q^{8} +(-\)\(21\!\cdots\!27\)\( - 127545655582776312 \beta) q^{9} +(\)\(16\!\cdots\!00\)\( - 10522244759337041920 \beta) q^{10} +(\)\(39\!\cdots\!72\)\( - \)\(23\!\cdots\!37\)\( \beta) q^{11} +(\)\(10\!\cdots\!36\)\( - \)\(20\!\cdots\!76\)\( \beta) q^{12} +(\)\(14\!\cdots\!86\)\( - \)\(19\!\cdots\!72\)\( \beta) q^{13} +(-\)\(58\!\cdots\!52\)\( + \)\(79\!\cdots\!12\)\( \beta) q^{14} +(-\)\(19\!\cdots\!00\)\( + \)\(55\!\cdots\!70\)\( \beta) q^{15} +\)\(34\!\cdots\!56\)\( q^{16} +(\)\(19\!\cdots\!82\)\( - \)\(10\!\cdots\!08\)\( \beta) q^{17} +(\)\(90\!\cdots\!92\)\( + \)\(54\!\cdots\!52\)\( \beta) q^{18} +(-\)\(69\!\cdots\!60\)\( + \)\(22\!\cdots\!53\)\( \beta) q^{19} +(-\)\(68\!\cdots\!00\)\( + \)\(45\!\cdots\!20\)\( \beta) q^{20} +(\)\(20\!\cdots\!52\)\( - \)\(16\!\cdots\!44\)\( \beta) q^{21} +(-\)\(16\!\cdots\!12\)\( + \)\(99\!\cdots\!52\)\( \beta) q^{22} +(\)\(71\!\cdots\!56\)\( - \)\(19\!\cdots\!26\)\( \beta) q^{23} +(-\)\(45\!\cdots\!56\)\( + \)\(87\!\cdots\!96\)\( \beta) q^{24} +(\)\(25\!\cdots\!75\)\( - \)\(18\!\cdots\!00\)\( \beta) q^{25} +(-\)\(61\!\cdots\!56\)\( + \)\(81\!\cdots\!12\)\( \beta) q^{26} +(\)\(18\!\cdots\!80\)\( + \)\(13\!\cdots\!18\)\( \beta) q^{27} +(\)\(24\!\cdots\!92\)\( - \)\(34\!\cdots\!52\)\( \beta) q^{28} +(\)\(31\!\cdots\!70\)\( + \)\(65\!\cdots\!76\)\( \beta) q^{29} +(\)\(83\!\cdots\!00\)\( - \)\(23\!\cdots\!20\)\( \beta) q^{30} +(\)\(15\!\cdots\!92\)\( + \)\(58\!\cdots\!76\)\( \beta) q^{31} -\)\(14\!\cdots\!76\)\( q^{32} +(\)\(18\!\cdots\!12\)\( - \)\(56\!\cdots\!44\)\( \beta) q^{33} +(-\)\(82\!\cdots\!72\)\( + \)\(46\!\cdots\!68\)\( \beta) q^{34} +(-\)\(79\!\cdots\!00\)\( + \)\(40\!\cdots\!40\)\( \beta) q^{35} +(-\)\(38\!\cdots\!32\)\( - \)\(23\!\cdots\!92\)\( \beta) q^{36} +(-\)\(10\!\cdots\!98\)\( + \)\(27\!\cdots\!52\)\( \beta) q^{37} +(\)\(29\!\cdots\!60\)\( - \)\(95\!\cdots\!88\)\( \beta) q^{38} +(\)\(14\!\cdots\!56\)\( - \)\(26\!\cdots\!58\)\( \beta) q^{39} +(\)\(29\!\cdots\!00\)\( - \)\(19\!\cdots\!20\)\( \beta) q^{40} +(-\)\(46\!\cdots\!38\)\( + \)\(95\!\cdots\!64\)\( \beta) q^{41} +(-\)\(89\!\cdots\!92\)\( + \)\(69\!\cdots\!24\)\( \beta) q^{42} +(\)\(75\!\cdots\!36\)\( - \)\(67\!\cdots\!73\)\( \beta) q^{43} +(\)\(72\!\cdots\!52\)\( - \)\(42\!\cdots\!92\)\( \beta) q^{44} +(-\)\(12\!\cdots\!50\)\( - \)\(40\!\cdots\!40\)\( \beta) q^{45} +(-\)\(30\!\cdots\!76\)\( + \)\(84\!\cdots\!96\)\( \beta) q^{46} +(-\)\(15\!\cdots\!48\)\( - \)\(17\!\cdots\!92\)\( \beta) q^{47} +(\)\(19\!\cdots\!76\)\( - \)\(37\!\cdots\!16\)\( \beta) q^{48} +(-\)\(64\!\cdots\!63\)\( - \)\(50\!\cdots\!28\)\( \beta) q^{49} +(-\)\(10\!\cdots\!00\)\( + \)\(78\!\cdots\!00\)\( \beta) q^{50} +(\)\(87\!\cdots\!72\)\( - \)\(27\!\cdots\!70\)\( \beta) q^{51} +(\)\(26\!\cdots\!76\)\( - \)\(35\!\cdots\!52\)\( \beta) q^{52} +(-\)\(26\!\cdots\!94\)\( - \)\(32\!\cdots\!72\)\( \beta) q^{53} +(-\)\(81\!\cdots\!80\)\( - \)\(56\!\cdots\!28\)\( \beta) q^{54} +(-\)\(50\!\cdots\!00\)\( + \)\(18\!\cdots\!90\)\( \beta) q^{55} +(-\)\(10\!\cdots\!32\)\( + \)\(14\!\cdots\!92\)\( \beta) q^{56} +(-\)\(16\!\cdots\!60\)\( + \)\(20\!\cdots\!48\)\( \beta) q^{57} +(-\)\(13\!\cdots\!20\)\( - \)\(28\!\cdots\!96\)\( \beta) q^{58} +(-\)\(32\!\cdots\!60\)\( - \)\(11\!\cdots\!93\)\( \beta) q^{59} +(-\)\(35\!\cdots\!00\)\( + \)\(10\!\cdots\!20\)\( \beta) q^{60} +(-\)\(14\!\cdots\!18\)\( + \)\(21\!\cdots\!72\)\( \beta) q^{61} +(-\)\(64\!\cdots\!32\)\( - \)\(25\!\cdots\!96\)\( \beta) q^{62} +(-\)\(13\!\cdots\!24\)\( - \)\(13\!\cdots\!50\)\( \beta) q^{63} +\)\(62\!\cdots\!96\)\( q^{64} +(-\)\(35\!\cdots\!00\)\( + \)\(10\!\cdots\!20\)\( \beta) q^{65} +(-\)\(80\!\cdots\!52\)\( + \)\(24\!\cdots\!24\)\( \beta) q^{66} +(-\)\(82\!\cdots\!88\)\( - \)\(39\!\cdots\!51\)\( \beta) q^{67} +(\)\(35\!\cdots\!12\)\( - \)\(20\!\cdots\!28\)\( \beta) q^{68} +(\)\(17\!\cdots\!76\)\( - \)\(90\!\cdots\!12\)\( \beta) q^{69} +(\)\(34\!\cdots\!00\)\( - \)\(17\!\cdots\!40\)\( \beta) q^{70} +(\)\(12\!\cdots\!32\)\( + \)\(52\!\cdots\!22\)\( \beta) q^{71} +(\)\(16\!\cdots\!72\)\( + \)\(10\!\cdots\!32\)\( \beta) q^{72} +(\)\(14\!\cdots\!66\)\( + \)\(11\!\cdots\!88\)\( \beta) q^{73} +(\)\(46\!\cdots\!08\)\( - \)\(11\!\cdots\!92\)\( \beta) q^{74} +(\)\(14\!\cdots\!00\)\( - \)\(38\!\cdots\!25\)\( \beta) q^{75} +(-\)\(12\!\cdots\!60\)\( + \)\(40\!\cdots\!48\)\( \beta) q^{76} +(\)\(80\!\cdots\!64\)\( - \)\(38\!\cdots\!28\)\( \beta) q^{77} +(-\)\(61\!\cdots\!76\)\( + \)\(11\!\cdots\!68\)\( \beta) q^{78} +(-\)\(36\!\cdots\!00\)\( + \)\(48\!\cdots\!68\)\( \beta) q^{79} +(-\)\(12\!\cdots\!00\)\( + \)\(83\!\cdots\!20\)\( \beta) q^{80} +(-\)\(69\!\cdots\!59\)\( + \)\(18\!\cdots\!64\)\( \beta) q^{81} +(\)\(19\!\cdots\!48\)\( - \)\(41\!\cdots\!44\)\( \beta) q^{82} +(-\)\(10\!\cdots\!24\)\( - \)\(15\!\cdots\!55\)\( \beta) q^{83} +(\)\(38\!\cdots\!32\)\( - \)\(29\!\cdots\!04\)\( \beta) q^{84} +(-\)\(24\!\cdots\!00\)\( + \)\(87\!\cdots\!40\)\( \beta) q^{85} +(-\)\(32\!\cdots\!56\)\( + \)\(29\!\cdots\!08\)\( \beta) q^{86} +(-\)\(27\!\cdots\!80\)\( - \)\(31\!\cdots\!74\)\( \beta) q^{87} +(-\)\(30\!\cdots\!92\)\( + \)\(18\!\cdots\!32\)\( \beta) q^{88} +(-\)\(13\!\cdots\!10\)\( + \)\(56\!\cdots\!40\)\( \beta) q^{89} +(\)\(51\!\cdots\!00\)\( + \)\(17\!\cdots\!40\)\( \beta) q^{90} +(\)\(42\!\cdots\!32\)\( - \)\(28\!\cdots\!56\)\( \beta) q^{91} +(\)\(13\!\cdots\!96\)\( - \)\(36\!\cdots\!16\)\( \beta) q^{92} +(-\)\(32\!\cdots\!68\)\( - \)\(13\!\cdots\!16\)\( \beta) q^{93} +(\)\(66\!\cdots\!08\)\( + \)\(74\!\cdots\!32\)\( \beta) q^{94} +(\)\(37\!\cdots\!00\)\( - \)\(10\!\cdots\!50\)\( \beta) q^{95} +(-\)\(83\!\cdots\!96\)\( + \)\(16\!\cdots\!36\)\( \beta) q^{96} +(-\)\(29\!\cdots\!18\)\( + \)\(25\!\cdots\!20\)\( \beta) q^{97} +(\)\(27\!\cdots\!48\)\( + \)\(21\!\cdots\!88\)\( \beta) q^{98} +(\)\(10\!\cdots\!56\)\( - \)\(74\!\cdots\!65\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8589934592q^{2} + 1149059960205192q^{3} + 36893488147419103232q^{4} - \)\(74\!\cdots\!00\)\(q^{5} - \)\(49\!\cdots\!32\)\(q^{6} + \)\(27\!\cdots\!24\)\(q^{7} - \)\(15\!\cdots\!72\)\(q^{8} - \)\(42\!\cdots\!54\)\(q^{9} + O(q^{10}) \) \( 2q - 8589934592q^{2} + 1149059960205192q^{3} + 36893488147419103232q^{4} - \)\(74\!\cdots\!00\)\(q^{5} - \)\(49\!\cdots\!32\)\(q^{6} + \)\(27\!\cdots\!24\)\(q^{7} - \)\(15\!\cdots\!72\)\(q^{8} - \)\(42\!\cdots\!54\)\(q^{9} + \)\(32\!\cdots\!00\)\(q^{10} + \)\(78\!\cdots\!44\)\(q^{11} + \)\(21\!\cdots\!72\)\(q^{12} + \)\(28\!\cdots\!72\)\(q^{13} - \)\(11\!\cdots\!04\)\(q^{14} - \)\(38\!\cdots\!00\)\(q^{15} + \)\(68\!\cdots\!12\)\(q^{16} + \)\(38\!\cdots\!64\)\(q^{17} + \)\(18\!\cdots\!84\)\(q^{18} - \)\(13\!\cdots\!20\)\(q^{19} - \)\(13\!\cdots\!00\)\(q^{20} + \)\(41\!\cdots\!04\)\(q^{21} - \)\(33\!\cdots\!24\)\(q^{22} + \)\(14\!\cdots\!12\)\(q^{23} - \)\(91\!\cdots\!12\)\(q^{24} + \)\(50\!\cdots\!50\)\(q^{25} - \)\(12\!\cdots\!12\)\(q^{26} + \)\(37\!\cdots\!60\)\(q^{27} + \)\(49\!\cdots\!84\)\(q^{28} + \)\(63\!\cdots\!40\)\(q^{29} + \)\(16\!\cdots\!00\)\(q^{30} + \)\(30\!\cdots\!84\)\(q^{31} - \)\(29\!\cdots\!52\)\(q^{32} + \)\(37\!\cdots\!24\)\(q^{33} - \)\(16\!\cdots\!44\)\(q^{34} - \)\(15\!\cdots\!00\)\(q^{35} - \)\(77\!\cdots\!64\)\(q^{36} - \)\(21\!\cdots\!96\)\(q^{37} + \)\(59\!\cdots\!20\)\(q^{38} + \)\(28\!\cdots\!12\)\(q^{39} + \)\(59\!\cdots\!00\)\(q^{40} - \)\(93\!\cdots\!76\)\(q^{41} - \)\(17\!\cdots\!84\)\(q^{42} + \)\(15\!\cdots\!72\)\(q^{43} + \)\(14\!\cdots\!04\)\(q^{44} - \)\(24\!\cdots\!00\)\(q^{45} - \)\(60\!\cdots\!52\)\(q^{46} - \)\(30\!\cdots\!96\)\(q^{47} + \)\(39\!\cdots\!52\)\(q^{48} - \)\(12\!\cdots\!26\)\(q^{49} - \)\(21\!\cdots\!00\)\(q^{50} + \)\(17\!\cdots\!44\)\(q^{51} + \)\(53\!\cdots\!52\)\(q^{52} - \)\(52\!\cdots\!88\)\(q^{53} - \)\(16\!\cdots\!60\)\(q^{54} - \)\(10\!\cdots\!00\)\(q^{55} - \)\(21\!\cdots\!64\)\(q^{56} - \)\(32\!\cdots\!20\)\(q^{57} - \)\(27\!\cdots\!40\)\(q^{58} - \)\(65\!\cdots\!20\)\(q^{59} - \)\(71\!\cdots\!00\)\(q^{60} - \)\(29\!\cdots\!36\)\(q^{61} - \)\(12\!\cdots\!64\)\(q^{62} - \)\(27\!\cdots\!48\)\(q^{63} + \)\(12\!\cdots\!92\)\(q^{64} - \)\(70\!\cdots\!00\)\(q^{65} - \)\(16\!\cdots\!04\)\(q^{66} - \)\(16\!\cdots\!76\)\(q^{67} + \)\(70\!\cdots\!24\)\(q^{68} + \)\(35\!\cdots\!52\)\(q^{69} + \)\(68\!\cdots\!00\)\(q^{70} + \)\(24\!\cdots\!64\)\(q^{71} + \)\(33\!\cdots\!44\)\(q^{72} + \)\(29\!\cdots\!32\)\(q^{73} + \)\(93\!\cdots\!16\)\(q^{74} + \)\(28\!\cdots\!00\)\(q^{75} - \)\(25\!\cdots\!20\)\(q^{76} + \)\(16\!\cdots\!28\)\(q^{77} - \)\(12\!\cdots\!52\)\(q^{78} - \)\(72\!\cdots\!00\)\(q^{79} - \)\(25\!\cdots\!00\)\(q^{80} - \)\(13\!\cdots\!18\)\(q^{81} + \)\(39\!\cdots\!96\)\(q^{82} - \)\(21\!\cdots\!48\)\(q^{83} + \)\(77\!\cdots\!64\)\(q^{84} - \)\(48\!\cdots\!00\)\(q^{85} - \)\(65\!\cdots\!12\)\(q^{86} - \)\(55\!\cdots\!60\)\(q^{87} - \)\(61\!\cdots\!84\)\(q^{88} - \)\(26\!\cdots\!20\)\(q^{89} + \)\(10\!\cdots\!00\)\(q^{90} + \)\(84\!\cdots\!64\)\(q^{91} + \)\(26\!\cdots\!92\)\(q^{92} - \)\(65\!\cdots\!36\)\(q^{93} + \)\(13\!\cdots\!16\)\(q^{94} + \)\(74\!\cdots\!00\)\(q^{95} - \)\(16\!\cdots\!92\)\(q^{96} - \)\(59\!\cdots\!36\)\(q^{97} + \)\(55\!\cdots\!96\)\(q^{98} + \)\(21\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

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
4.42861e7
−4.42861e7
−4.29497e9 −2.22863e15 1.84467e19 2.44878e22 9.57189e24 8.86741e26 −7.92282e28 −5.33426e30 −1.05174e32
1.2 −4.29497e9 3.37769e15 1.84467e19 −9.92503e22 −1.45071e25 1.82190e27 −7.92282e28 1.10774e30 4.26277e32
\(n\): 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 \( T_{3}^{2} - \)\(11\!\cdots\!92\)\( T_{3} - \)\(75\!\cdots\!84\)\( \) acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(2))\).