Properties

Label 2.48.a.b
Level 2
Weight 48
Character orbit 2.a
Self dual Yes
Analytic conductor 27.982
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 48 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(27.981532531\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{3}\cdot 5 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8640\sqrt{23589383914321}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -8388608 q^{2} + ( 61144922412 - 5 \beta ) q^{3} + 70368744177664 q^{4} + ( 9089248919597070 + 460908 \beta ) q^{5} + ( -512920785304682496 + 41943040 \beta ) q^{6} + ( 80509272728013413816 - 675210690 \beta ) q^{7} -590295810358705651712 q^{8} + ( 21173339014074419649957 - 611449224120 \beta ) q^{9} +O(q^{10})\) \( q -8388608 q^{2} +(61144922412 - 5 \beta) q^{3} +70368744177664 q^{4} +(9089248919597070 + 460908 \beta) q^{5} +(-512920785304682496 + 41943040 \beta) q^{6} +(80509272728013413816 - 675210690 \beta) q^{7} -\)\(59\!\cdots\!12\)\( q^{8} +(\)\(21\!\cdots\!57\)\( - 611449224120 \beta) q^{9} +(-\)\(76\!\cdots\!60\)\( - 3866376536064 \beta) q^{10} +(\)\(57\!\cdots\!92\)\( - 79914943998855 \beta) q^{11} +(\)\(43\!\cdots\!68\)\( - 351843720888320 \beta) q^{12} +(-\)\(45\!\cdots\!18\)\( + 4479527100978540 \beta) q^{13} +(-\)\(67\!\cdots\!28\)\( + 5664077795819520 \beta) q^{14} +(-\)\(35\!\cdots\!60\)\( - 17264060698915254 \beta) q^{15} +\)\(49\!\cdots\!96\)\( q^{16} +(-\)\(51\!\cdots\!94\)\( - 2192118261102219960 \beta) q^{17} +(-\)\(17\!\cdots\!56\)\( + 5129207853046824960 \beta) q^{18} +(\)\(51\!\cdots\!60\)\( + 15893727992186364855 \beta) q^{19} +(\)\(63\!\cdots\!80\)\( + 32433517141438758912 \beta) q^{20} +(\)\(10\!\cdots\!92\)\( - \)\(44\!\cdots\!60\)\( \beta) q^{21} +(-\)\(48\!\cdots\!36\)\( + \)\(67\!\cdots\!40\)\( \beta) q^{22} +(\)\(43\!\cdots\!12\)\( - \)\(68\!\cdots\!30\)\( \beta) q^{23} +(-\)\(36\!\cdots\!44\)\( + \)\(29\!\cdots\!60\)\( \beta) q^{24} +(-\)\(25\!\cdots\!25\)\( + \)\(83\!\cdots\!20\)\( \beta) q^{25} +(\)\(38\!\cdots\!44\)\( - \)\(37\!\cdots\!20\)\( \beta) q^{26} +(\)\(50\!\cdots\!40\)\( - \)\(10\!\cdots\!90\)\( \beta) q^{27} +(\)\(56\!\cdots\!24\)\( - \)\(47\!\cdots\!60\)\( \beta) q^{28} +(\)\(22\!\cdots\!30\)\( + \)\(51\!\cdots\!60\)\( \beta) q^{29} +(\)\(29\!\cdots\!80\)\( + \)\(14\!\cdots\!32\)\( \beta) q^{30} +(\)\(10\!\cdots\!52\)\( - \)\(16\!\cdots\!60\)\( \beta) q^{31} -\)\(41\!\cdots\!68\)\( q^{32} +(\)\(73\!\cdots\!04\)\( - \)\(77\!\cdots\!20\)\( \beta) q^{33} +(\)\(43\!\cdots\!52\)\( + \)\(18\!\cdots\!80\)\( \beta) q^{34} +(\)\(18\!\cdots\!20\)\( + \)\(30\!\cdots\!28\)\( \beta) q^{35} +(\)\(14\!\cdots\!48\)\( - \)\(43\!\cdots\!80\)\( \beta) q^{36} +(\)\(77\!\cdots\!26\)\( - \)\(74\!\cdots\!60\)\( \beta) q^{37} +(-\)\(42\!\cdots\!80\)\( - \)\(13\!\cdots\!40\)\( \beta) q^{38} +(-\)\(42\!\cdots\!16\)\( + \)\(50\!\cdots\!70\)\( \beta) q^{39} +(-\)\(53\!\cdots\!40\)\( - \)\(27\!\cdots\!96\)\( \beta) q^{40} +(-\)\(91\!\cdots\!58\)\( - \)\(29\!\cdots\!40\)\( \beta) q^{41} +(-\)\(91\!\cdots\!36\)\( + \)\(37\!\cdots\!80\)\( \beta) q^{42} +(\)\(12\!\cdots\!52\)\( - \)\(58\!\cdots\!15\)\( \beta) q^{43} +(\)\(40\!\cdots\!88\)\( - \)\(56\!\cdots\!20\)\( \beta) q^{44} +(-\)\(30\!\cdots\!10\)\( + \)\(42\!\cdots\!56\)\( \beta) q^{45} +(-\)\(36\!\cdots\!96\)\( + \)\(57\!\cdots\!40\)\( \beta) q^{46} +(\)\(14\!\cdots\!56\)\( + \)\(49\!\cdots\!60\)\( \beta) q^{47} +(\)\(30\!\cdots\!52\)\( - \)\(24\!\cdots\!80\)\( \beta) q^{48} +(\)\(20\!\cdots\!13\)\( - \)\(10\!\cdots\!80\)\( \beta) q^{49} +(\)\(21\!\cdots\!00\)\( - \)\(70\!\cdots\!60\)\( \beta) q^{50} +(\)\(16\!\cdots\!72\)\( + \)\(12\!\cdots\!50\)\( \beta) q^{51} +(-\)\(32\!\cdots\!52\)\( + \)\(31\!\cdots\!60\)\( \beta) q^{52} +(\)\(18\!\cdots\!82\)\( - \)\(12\!\cdots\!60\)\( \beta) q^{53} +(-\)\(42\!\cdots\!20\)\( + \)\(86\!\cdots\!20\)\( \beta) q^{54} +(-\)\(59\!\cdots\!60\)\( - \)\(46\!\cdots\!14\)\( \beta) q^{55} +(-\)\(47\!\cdots\!92\)\( + \)\(39\!\cdots\!80\)\( \beta) q^{56} +(-\)\(10\!\cdots\!80\)\( - \)\(15\!\cdots\!40\)\( \beta) q^{57} +(-\)\(18\!\cdots\!40\)\( - \)\(42\!\cdots\!80\)\( \beta) q^{58} +(\)\(10\!\cdots\!60\)\( + \)\(11\!\cdots\!45\)\( \beta) q^{59} +(-\)\(24\!\cdots\!40\)\( - \)\(12\!\cdots\!56\)\( \beta) q^{60} +(\)\(11\!\cdots\!02\)\( + \)\(22\!\cdots\!80\)\( \beta) q^{61} +(-\)\(87\!\cdots\!16\)\( + \)\(13\!\cdots\!80\)\( \beta) q^{62} +(\)\(24\!\cdots\!12\)\( - \)\(63\!\cdots\!50\)\( \beta) q^{63} +\)\(34\!\cdots\!44\)\( q^{64} +(\)\(32\!\cdots\!40\)\( + \)\(19\!\cdots\!56\)\( \beta) q^{65} +(-\)\(61\!\cdots\!32\)\( + \)\(65\!\cdots\!60\)\( \beta) q^{66} +(\)\(48\!\cdots\!96\)\( + \)\(15\!\cdots\!55\)\( \beta) q^{67} +(-\)\(36\!\cdots\!16\)\( - \)\(15\!\cdots\!40\)\( \beta) q^{68} +(\)\(63\!\cdots\!44\)\( - \)\(64\!\cdots\!20\)\( \beta) q^{69} +(-\)\(15\!\cdots\!60\)\( - \)\(25\!\cdots\!24\)\( \beta) q^{70} +(-\)\(17\!\cdots\!28\)\( - \)\(16\!\cdots\!70\)\( \beta) q^{71} +(-\)\(12\!\cdots\!84\)\( + \)\(36\!\cdots\!40\)\( \beta) q^{72} +(-\)\(22\!\cdots\!18\)\( - \)\(39\!\cdots\!60\)\( \beta) q^{73} +(-\)\(64\!\cdots\!08\)\( + \)\(62\!\cdots\!80\)\( \beta) q^{74} +(-\)\(89\!\cdots\!00\)\( + \)\(17\!\cdots\!65\)\( \beta) q^{75} +(\)\(35\!\cdots\!40\)\( + \)\(11\!\cdots\!20\)\( \beta) q^{76} +(\)\(14\!\cdots\!72\)\( - \)\(68\!\cdots\!60\)\( \beta) q^{77} +(\)\(35\!\cdots\!28\)\( - \)\(42\!\cdots\!60\)\( \beta) q^{78} +(\)\(22\!\cdots\!00\)\( + \)\(97\!\cdots\!80\)\( \beta) q^{79} +(\)\(45\!\cdots\!20\)\( + \)\(22\!\cdots\!68\)\( \beta) q^{80} +(-\)\(16\!\cdots\!79\)\( - \)\(96\!\cdots\!40\)\( \beta) q^{81} +(\)\(77\!\cdots\!64\)\( + \)\(24\!\cdots\!20\)\( \beta) q^{82} +(-\)\(67\!\cdots\!08\)\( + \)\(40\!\cdots\!75\)\( \beta) q^{83} +(\)\(76\!\cdots\!88\)\( - \)\(31\!\cdots\!40\)\( \beta) q^{84} +(-\)\(22\!\cdots\!80\)\( - \)\(43\!\cdots\!52\)\( \beta) q^{85} +(-\)\(10\!\cdots\!16\)\( + \)\(48\!\cdots\!20\)\( \beta) q^{86} +(-\)\(31\!\cdots\!40\)\( - \)\(80\!\cdots\!30\)\( \beta) q^{87} +(-\)\(33\!\cdots\!04\)\( + \)\(47\!\cdots\!60\)\( \beta) q^{88} +(\)\(22\!\cdots\!10\)\( + \)\(28\!\cdots\!00\)\( \beta) q^{89} +(\)\(25\!\cdots\!80\)\( - \)\(35\!\cdots\!48\)\( \beta) q^{90} +(-\)\(89\!\cdots\!88\)\( + \)\(39\!\cdots\!60\)\( \beta) q^{91} +(\)\(30\!\cdots\!68\)\( - \)\(48\!\cdots\!20\)\( \beta) q^{92} +(\)\(20\!\cdots\!24\)\( - \)\(61\!\cdots\!80\)\( \beta) q^{93} +(-\)\(12\!\cdots\!48\)\( - \)\(41\!\cdots\!80\)\( \beta) q^{94} +(\)\(17\!\cdots\!00\)\( + \)\(37\!\cdots\!30\)\( \beta) q^{95} +(-\)\(25\!\cdots\!16\)\( + \)\(20\!\cdots\!40\)\( \beta) q^{96} +(\)\(26\!\cdots\!46\)\( + \)\(78\!\cdots\!00\)\( \beta) q^{97} +(-\)\(17\!\cdots\!04\)\( + \)\(91\!\cdots\!40\)\( \beta) q^{98} +(\)\(98\!\cdots\!44\)\( - \)\(20\!\cdots\!75\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16777216q^{2} + 122289844824q^{3} + 140737488355328q^{4} + 18178497839194140q^{5} - 1025841570609364992q^{6} + 161018545456026827632q^{7} - 1180591620717411303424q^{8} + 42346678028148839299914q^{9} + O(q^{10}) \) \( 2q - 16777216q^{2} + 122289844824q^{3} + 140737488355328q^{4} + 18178497839194140q^{5} - 1025841570609364992q^{6} + \)\(16\!\cdots\!32\)\(q^{7} - \)\(11\!\cdots\!24\)\(q^{8} + \)\(42\!\cdots\!14\)\(q^{9} - \)\(15\!\cdots\!20\)\(q^{10} + \)\(11\!\cdots\!84\)\(q^{11} + \)\(86\!\cdots\!36\)\(q^{12} - \)\(91\!\cdots\!36\)\(q^{13} - \)\(13\!\cdots\!56\)\(q^{14} - \)\(70\!\cdots\!20\)\(q^{15} + \)\(99\!\cdots\!92\)\(q^{16} - \)\(10\!\cdots\!88\)\(q^{17} - \)\(35\!\cdots\!12\)\(q^{18} + \)\(10\!\cdots\!20\)\(q^{19} + \)\(12\!\cdots\!60\)\(q^{20} + \)\(21\!\cdots\!84\)\(q^{21} - \)\(96\!\cdots\!72\)\(q^{22} + \)\(87\!\cdots\!24\)\(q^{23} - \)\(72\!\cdots\!88\)\(q^{24} - \)\(50\!\cdots\!50\)\(q^{25} + \)\(76\!\cdots\!88\)\(q^{26} + \)\(10\!\cdots\!80\)\(q^{27} + \)\(11\!\cdots\!48\)\(q^{28} + \)\(44\!\cdots\!60\)\(q^{29} + \)\(58\!\cdots\!60\)\(q^{30} + \)\(20\!\cdots\!04\)\(q^{31} - \)\(83\!\cdots\!36\)\(q^{32} + \)\(14\!\cdots\!08\)\(q^{33} + \)\(86\!\cdots\!04\)\(q^{34} + \)\(36\!\cdots\!40\)\(q^{35} + \)\(29\!\cdots\!96\)\(q^{36} + \)\(15\!\cdots\!52\)\(q^{37} - \)\(85\!\cdots\!60\)\(q^{38} - \)\(84\!\cdots\!32\)\(q^{39} - \)\(10\!\cdots\!80\)\(q^{40} - \)\(18\!\cdots\!16\)\(q^{41} - \)\(18\!\cdots\!72\)\(q^{42} + \)\(25\!\cdots\!04\)\(q^{43} + \)\(80\!\cdots\!76\)\(q^{44} - \)\(60\!\cdots\!20\)\(q^{45} - \)\(73\!\cdots\!92\)\(q^{46} + \)\(28\!\cdots\!12\)\(q^{47} + \)\(60\!\cdots\!04\)\(q^{48} + \)\(40\!\cdots\!26\)\(q^{49} + \)\(42\!\cdots\!00\)\(q^{50} + \)\(32\!\cdots\!44\)\(q^{51} - \)\(64\!\cdots\!04\)\(q^{52} + \)\(36\!\cdots\!64\)\(q^{53} - \)\(84\!\cdots\!40\)\(q^{54} - \)\(11\!\cdots\!20\)\(q^{55} - \)\(95\!\cdots\!84\)\(q^{56} - \)\(21\!\cdots\!60\)\(q^{57} - \)\(37\!\cdots\!80\)\(q^{58} + \)\(21\!\cdots\!20\)\(q^{59} - \)\(49\!\cdots\!80\)\(q^{60} + \)\(23\!\cdots\!04\)\(q^{61} - \)\(17\!\cdots\!32\)\(q^{62} + \)\(48\!\cdots\!24\)\(q^{63} + \)\(69\!\cdots\!88\)\(q^{64} + \)\(64\!\cdots\!80\)\(q^{65} - \)\(12\!\cdots\!64\)\(q^{66} + \)\(96\!\cdots\!92\)\(q^{67} - \)\(72\!\cdots\!32\)\(q^{68} + \)\(12\!\cdots\!88\)\(q^{69} - \)\(30\!\cdots\!20\)\(q^{70} - \)\(34\!\cdots\!56\)\(q^{71} - \)\(24\!\cdots\!68\)\(q^{72} - \)\(44\!\cdots\!36\)\(q^{73} - \)\(12\!\cdots\!16\)\(q^{74} - \)\(17\!\cdots\!00\)\(q^{75} + \)\(71\!\cdots\!80\)\(q^{76} + \)\(28\!\cdots\!44\)\(q^{77} + \)\(70\!\cdots\!56\)\(q^{78} + \)\(45\!\cdots\!00\)\(q^{79} + \)\(90\!\cdots\!40\)\(q^{80} - \)\(32\!\cdots\!58\)\(q^{81} + \)\(15\!\cdots\!28\)\(q^{82} - \)\(13\!\cdots\!16\)\(q^{83} + \)\(15\!\cdots\!76\)\(q^{84} - \)\(44\!\cdots\!60\)\(q^{85} - \)\(21\!\cdots\!32\)\(q^{86} - \)\(62\!\cdots\!80\)\(q^{87} - \)\(67\!\cdots\!08\)\(q^{88} + \)\(45\!\cdots\!20\)\(q^{89} + \)\(50\!\cdots\!60\)\(q^{90} - \)\(17\!\cdots\!76\)\(q^{91} + \)\(61\!\cdots\!36\)\(q^{92} + \)\(40\!\cdots\!48\)\(q^{93} - \)\(24\!\cdots\!96\)\(q^{94} + \)\(35\!\cdots\!00\)\(q^{95} - \)\(50\!\cdots\!32\)\(q^{96} + \)\(53\!\cdots\!92\)\(q^{97} - \)\(34\!\cdots\!08\)\(q^{98} + \)\(19\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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
2.42845e6
−2.42844e6
−8.38861e6 −1.48673e11 7.03687e13 2.84306e16 1.24716e18 5.21750e19 −5.90296e20 −4.48523e21 −2.38493e23
1.2 −8.38861e6 2.70963e11 7.03687e13 −1.02521e16 −2.27300e18 1.08843e20 −5.90296e20 4.68319e22 8.60007e22
\(n\): 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 \( T_{3}^{2} - 122289844824 T_{3} - \)\(40\!\cdots\!56\)\( \) acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(2))\).