Properties

Label 2.50.a.a
Level 2
Weight 50
Character orbit 2.a
Self dual Yes
Analytic conductor 30.413
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 50 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(30.4132410198\)
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: \( 2^{7}\cdot 3^{3}\cdot 5^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 43200\sqrt{88518163840129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -16777216 q^{2} \) \( + ( 140525537796 - \beta ) q^{3} \) \( + 281474976710656 q^{4} \) \( + ( 41587265591271750 - 349020 \beta ) q^{5} \) \( + ( -2357627301119655936 + 16777216 \beta ) q^{6} \) \( + ( -216194632642999109368 + 1176353838 \beta ) q^{7} \) \( -4722366482869645213696 q^{8} \) \( + ( -54355764372760160092467 - 281051075592 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(-16777216 q^{2}\) \(+(140525537796 - \beta) q^{3}\) \(+281474976710656 q^{4}\) \(+(41587265591271750 - 349020 \beta) q^{5}\) \(+(-2357627301119655936 + 16777216 \beta) q^{6}\) \(+(-\)\(21\!\cdots\!68\)\( + 1176353838 \beta) q^{7}\) \(-\)\(47\!\cdots\!96\)\( q^{8}\) \(+(-\)\(54\!\cdots\!67\)\( - 281051075592 \beta) q^{9}\) \(+(-\)\(69\!\cdots\!00\)\( + 5855583928320 \beta) q^{10}\) \(+(-\)\(12\!\cdots\!68\)\( + 91137694171293 \beta) q^{11}\) \(+(\)\(39\!\cdots\!76\)\( - 281474976710656 \beta) q^{12}\) \(+(-\)\(83\!\cdots\!14\)\( + 4871108754480708 \beta) q^{13}\) \(+(\)\(36\!\cdots\!88\)\( - 19735942432555008 \beta) q^{14}\) \(+(\)\(63\!\cdots\!00\)\( - 90633488792831670 \beta) q^{15}\) \(+\)\(79\!\cdots\!36\)\( q^{16}\) \(+(\)\(62\!\cdots\!82\)\( + 1739956737232606392 \beta) q^{17}\) \(+(\)\(91\!\cdots\!72\)\( + 4715254602239311872 \beta) q^{18}\) \(+(\)\(15\!\cdots\!60\)\( - 15619643210757982437 \beta) q^{19}\) \(+(\)\(11\!\cdots\!00\)\( - 98240396371553157120 \beta) q^{20}\) \(+(-\)\(22\!\cdots\!28\)\( + \)\(38\!\cdots\!16\)\( \beta) q^{21}\) \(+(\)\(21\!\cdots\!88\)\( - \)\(15\!\cdots\!88\)\( \beta) q^{22}\) \(+(-\)\(14\!\cdots\!64\)\( + \)\(49\!\cdots\!14\)\( \beta) q^{23}\) \(+(-\)\(66\!\cdots\!16\)\( + \)\(47\!\cdots\!96\)\( \beta) q^{24}\) \(+(\)\(40\!\cdots\!75\)\( - \)\(29\!\cdots\!00\)\( \beta) q^{25}\) \(+(\)\(14\!\cdots\!24\)\( - \)\(81\!\cdots\!28\)\( \beta) q^{26}\) \(+(\)\(51\!\cdots\!00\)\( + \)\(25\!\cdots\!18\)\( \beta) q^{27}\) \(+(-\)\(60\!\cdots\!08\)\( + \)\(33\!\cdots\!28\)\( \beta) q^{28}\) \(+(-\)\(59\!\cdots\!90\)\( - \)\(12\!\cdots\!04\)\( \beta) q^{29}\) \(+(-\)\(10\!\cdots\!00\)\( + \)\(15\!\cdots\!20\)\( \beta) q^{30}\) \(+(-\)\(35\!\cdots\!48\)\( - \)\(35\!\cdots\!44\)\( \beta) q^{31}\) \(-\)\(13\!\cdots\!76\)\( q^{32}\) \(+(-\)\(15\!\cdots\!28\)\( + \)\(14\!\cdots\!96\)\( \beta) q^{33}\) \(+(-\)\(10\!\cdots\!12\)\( - \)\(29\!\cdots\!72\)\( \beta) q^{34}\) \(+(-\)\(76\!\cdots\!00\)\( + \)\(12\!\cdots\!60\)\( \beta) q^{35}\) \(+(-\)\(15\!\cdots\!52\)\( - \)\(79\!\cdots\!52\)\( \beta) q^{36}\) \(+(-\)\(22\!\cdots\!58\)\( - \)\(60\!\cdots\!08\)\( \beta) q^{37}\) \(+(-\)\(26\!\cdots\!60\)\( + \)\(26\!\cdots\!92\)\( \beta) q^{38}\) \(+(-\)\(92\!\cdots\!44\)\( + \)\(15\!\cdots\!82\)\( \beta) q^{39}\) \(+(-\)\(19\!\cdots\!00\)\( + \)\(16\!\cdots\!20\)\( \beta) q^{40}\) \(+(\)\(34\!\cdots\!82\)\( - \)\(38\!\cdots\!76\)\( \beta) q^{41}\) \(+(\)\(37\!\cdots\!48\)\( - \)\(64\!\cdots\!56\)\( \beta) q^{42}\) \(+(\)\(11\!\cdots\!16\)\( - \)\(24\!\cdots\!63\)\( \beta) q^{43}\) \(+(-\)\(35\!\cdots\!08\)\( + \)\(25\!\cdots\!08\)\( \beta) q^{44}\) \(+(\)\(13\!\cdots\!50\)\( + \)\(72\!\cdots\!40\)\( \beta) q^{45}\) \(+(\)\(24\!\cdots\!24\)\( - \)\(82\!\cdots\!24\)\( \beta) q^{46}\) \(+(-\)\(59\!\cdots\!48\)\( + \)\(19\!\cdots\!48\)\( \beta) q^{47}\) \(+(\)\(11\!\cdots\!56\)\( - \)\(79\!\cdots\!36\)\( \beta) q^{48}\) \(+(\)\(18\!\cdots\!17\)\( - \)\(50\!\cdots\!68\)\( \beta) q^{49}\) \(+(-\)\(68\!\cdots\!00\)\( + \)\(48\!\cdots\!00\)\( \beta) q^{50}\) \(+(-\)\(20\!\cdots\!28\)\( - \)\(37\!\cdots\!50\)\( \beta) q^{51}\) \(+(-\)\(23\!\cdots\!84\)\( + \)\(13\!\cdots\!48\)\( \beta) q^{52}\) \(+(-\)\(19\!\cdots\!34\)\( + \)\(10\!\cdots\!28\)\( \beta) q^{53}\) \(+(-\)\(86\!\cdots\!00\)\( - \)\(42\!\cdots\!88\)\( \beta) q^{54}\) \(+(-\)\(53\!\cdots\!00\)\( + \)\(42\!\cdots\!10\)\( \beta) q^{55}\) \(+(\)\(10\!\cdots\!28\)\( - \)\(55\!\cdots\!48\)\( \beta) q^{56}\) \(+(\)\(48\!\cdots\!60\)\( - \)\(18\!\cdots\!12\)\( \beta) q^{57}\) \(+(\)\(99\!\cdots\!40\)\( + \)\(21\!\cdots\!64\)\( \beta) q^{58}\) \(+(\)\(10\!\cdots\!20\)\( + \)\(71\!\cdots\!77\)\( \beta) q^{59}\) \(+(\)\(17\!\cdots\!00\)\( - \)\(25\!\cdots\!20\)\( \beta) q^{60}\) \(+(\)\(28\!\cdots\!02\)\( - \)\(15\!\cdots\!48\)\( \beta) q^{61}\) \(+(\)\(59\!\cdots\!68\)\( + \)\(59\!\cdots\!04\)\( \beta) q^{62}\) \(+(-\)\(42\!\cdots\!44\)\( - \)\(31\!\cdots\!90\)\( \beta) q^{63}\) \(+\)\(22\!\cdots\!16\)\( q^{64}\) \(+(-\)\(31\!\cdots\!00\)\( + \)\(49\!\cdots\!80\)\( \beta) q^{65}\) \(+(\)\(25\!\cdots\!48\)\( - \)\(23\!\cdots\!36\)\( \beta) q^{66}\) \(+(-\)\(56\!\cdots\!08\)\( - \)\(89\!\cdots\!41\)\( \beta) q^{67}\) \(+(\)\(17\!\cdots\!92\)\( + \)\(48\!\cdots\!52\)\( \beta) q^{68}\) \(+(-\)\(10\!\cdots\!44\)\( + \)\(21\!\cdots\!08\)\( \beta) q^{69}\) \(+(\)\(12\!\cdots\!00\)\( - \)\(20\!\cdots\!60\)\( \beta) q^{70}\) \(+(-\)\(12\!\cdots\!08\)\( - \)\(42\!\cdots\!78\)\( \beta) q^{71}\) \(+(\)\(25\!\cdots\!32\)\( + \)\(13\!\cdots\!32\)\( \beta) q^{72}\) \(+(-\)\(48\!\cdots\!94\)\( + \)\(61\!\cdots\!28\)\( \beta) q^{73}\) \(+(\)\(37\!\cdots\!28\)\( + \)\(10\!\cdots\!28\)\( \beta) q^{74}\) \(+(\)\(53\!\cdots\!00\)\( - \)\(81\!\cdots\!75\)\( \beta) q^{75}\) \(+(\)\(44\!\cdots\!60\)\( - \)\(43\!\cdots\!72\)\( \beta) q^{76}\) \(+(\)\(17\!\cdots\!24\)\( - \)\(21\!\cdots\!08\)\( \beta) q^{77}\) \(+(\)\(15\!\cdots\!04\)\( - \)\(25\!\cdots\!12\)\( \beta) q^{78}\) \(+(-\)\(84\!\cdots\!80\)\( + \)\(65\!\cdots\!48\)\( \beta) q^{79}\) \(+(\)\(32\!\cdots\!00\)\( - \)\(27\!\cdots\!20\)\( \beta) q^{80}\) \(+(-\)\(28\!\cdots\!39\)\( + \)\(97\!\cdots\!64\)\( \beta) q^{81}\) \(+(-\)\(58\!\cdots\!12\)\( + \)\(63\!\cdots\!16\)\( \beta) q^{82}\) \(+(-\)\(12\!\cdots\!64\)\( - \)\(30\!\cdots\!25\)\( \beta) q^{83}\) \(+(-\)\(63\!\cdots\!68\)\( + \)\(10\!\cdots\!96\)\( \beta) q^{84}\) \(+(-\)\(74\!\cdots\!00\)\( - \)\(14\!\cdots\!40\)\( \beta) q^{85}\) \(+(-\)\(19\!\cdots\!56\)\( + \)\(40\!\cdots\!08\)\( \beta) q^{86}\) \(+(\)\(12\!\cdots\!60\)\( + \)\(41\!\cdots\!06\)\( \beta) q^{87}\) \(+(\)\(59\!\cdots\!28\)\( - \)\(43\!\cdots\!28\)\( \beta) q^{88}\) \(+(\)\(50\!\cdots\!90\)\( + \)\(84\!\cdots\!40\)\( \beta) q^{89}\) \(+(-\)\(23\!\cdots\!00\)\( - \)\(12\!\cdots\!40\)\( \beta) q^{90}\) \(+(\)\(11\!\cdots\!52\)\( - \)\(20\!\cdots\!76\)\( \beta) q^{91}\) \(+(-\)\(40\!\cdots\!84\)\( + \)\(13\!\cdots\!84\)\( \beta) q^{92}\) \(+(\)\(90\!\cdots\!92\)\( + \)\(30\!\cdots\!24\)\( \beta) q^{93}\) \(+(\)\(99\!\cdots\!68\)\( - \)\(32\!\cdots\!68\)\( \beta) q^{94}\) \(+(\)\(15\!\cdots\!00\)\( - \)\(61\!\cdots\!50\)\( \beta) q^{95}\) \(+(-\)\(18\!\cdots\!96\)\( + \)\(13\!\cdots\!76\)\( \beta) q^{96}\) \(+(\)\(43\!\cdots\!62\)\( + \)\(97\!\cdots\!40\)\( \beta) q^{97}\) \(+(-\)\(30\!\cdots\!72\)\( + \)\(85\!\cdots\!88\)\( \beta) q^{98}\) \(+(-\)\(41\!\cdots\!44\)\( - \)\(46\!\cdots\!75\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 33554432q^{2} \) \(\mathstrut +\mathstrut 281051075592q^{3} \) \(\mathstrut +\mathstrut 562949953421312q^{4} \) \(\mathstrut +\mathstrut 83174531182543500q^{5} \) \(\mathstrut -\mathstrut 4715254602239311872q^{6} \) \(\mathstrut -\mathstrut 432389265285998218736q^{7} \) \(\mathstrut -\mathstrut 9444732965739290427392q^{8} \) \(\mathstrut -\mathstrut 108711528745520320184934q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 33554432q^{2} \) \(\mathstrut +\mathstrut 281051075592q^{3} \) \(\mathstrut +\mathstrut 562949953421312q^{4} \) \(\mathstrut +\mathstrut 83174531182543500q^{5} \) \(\mathstrut -\mathstrut 4715254602239311872q^{6} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!36\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!92\)\(q^{8} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!34\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!36\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!52\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!28\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!76\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!72\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!64\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!44\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!20\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!56\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!76\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!28\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!32\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!48\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!16\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!80\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!96\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!52\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!56\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!24\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!04\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!16\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!88\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!64\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!96\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!32\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!16\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!48\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!96\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!12\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!34\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!56\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!68\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!68\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!56\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!80\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!40\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!04\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!36\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!88\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!32\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!96\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!16\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!84\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!88\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!16\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!64\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!88\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!56\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!20\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!48\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!08\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!60\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!78\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!24\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!28\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!36\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!12\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!20\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!56\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!04\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!68\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!84\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!36\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!92\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!24\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!44\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
4.70421e6
−4.70420e6
−1.67772e7 −2.65918e11 2.81475e14 −1.00270e17 4.46136e18 2.61926e20 −4.72237e21 −1.68587e23 1.68224e24
1.2 −1.67772e7 5.46969e11 2.81475e14 1.83444e17 −9.17661e18 −6.94316e20 −4.72237e21 5.98756e22 −3.07768e24
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 281051075592 T_{3} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!84\)\( \) acting on \(S_{50}^{\mathrm{new}}(\Gamma_0(2))\).