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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.

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

Display 100 coefficients 10 coefficients

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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 122289844824 T_{3} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!56\)\( \) acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(2))\).