Properties

Label 2.38.a.b
Level 2
Weight 38
Character orbit 2.a
Self dual Yes
Analytic conductor 17.343
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(17.3428076249\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 1920\sqrt{223572801841}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 262144 q^{2} + ( -250843404 - \beta ) q^{3} + 68719476736 q^{4} + ( 2091300429750 - 2700 \beta ) q^{5} + ( -65757093298176 - 262144 \beta ) q^{6} + ( -1756339478768728 - 5318082 \beta ) q^{7} + 18014398509481984 q^{8} + ( 436817284145972253 + 501686808 \beta ) q^{9} +O(q^{10})\) \( q +262144 q^{2} +(-250843404 - \beta) q^{3} +68719476736 q^{4} +(2091300429750 - 2700 \beta) q^{5} +(-65757093298176 - 262144 \beta) q^{6} +(-1756339478768728 - 5318082 \beta) q^{7} +18014398509481984 q^{8} +(436817284145972253 + 501686808 \beta) q^{9} +(548221859856384000 - 707788800 \beta) q^{10} +(12829972861280186652 - 7107001587 \beta) q^{11} +(-17237827465557049344 - 68719476736 \beta) q^{12} +(-75190417767106126114 + 695679299028 \beta) q^{13} +(-\)\(46\!\cdots\!32\)\( - 1394103287808 \beta) q^{14} +(\)\(17\!\cdots\!00\)\( - 1414023238950 \beta) q^{15} +\)\(47\!\cdots\!96\)\( q^{16} +(\)\(82\!\cdots\!82\)\( + 14083120550232 \beta) q^{17} +(\)\(11\!\cdots\!32\)\( + 131514186596352 \beta) q^{18} +(\)\(15\!\cdots\!00\)\( - 556602440642757 \beta) q^{19} +(\)\(14\!\cdots\!00\)\( - 185542587187200 \beta) q^{20} +(\)\(48\!\cdots\!12\)\( + 3090345270399856 \beta) q^{21} +(\)\(33\!\cdots\!88\)\( - 1863057824022528 \beta) q^{22} +(\)\(11\!\cdots\!96\)\( - 723601869769926 \beta) q^{23} +(-\)\(45\!\cdots\!36\)\( - 18014398509481984 \beta) q^{24} +(-\)\(62\!\cdots\!25\)\( - 11293022320650000 \beta) q^{25} +(-\)\(19\!\cdots\!16\)\( + 182368154164396032 \beta) q^{26} +(-\)\(41\!\cdots\!60\)\( - 112378204915589322 \beta) q^{27} +(-\)\(12\!\cdots\!08\)\( - 365455812279140352 \beta) q^{28} +(-\)\(77\!\cdots\!10\)\( + 99413651854516356 \beta) q^{29} +(\)\(44\!\cdots\!00\)\( - 370677707951308800 \beta) q^{30} +(\)\(24\!\cdots\!72\)\( + 3512250494601498936 \beta) q^{31} +\)\(12\!\cdots\!24\)\( q^{32} +(\)\(26\!\cdots\!92\)\( - 11047228390963704504 \beta) q^{33} +(\)\(21\!\cdots\!08\)\( + 3691805553520017408 \beta) q^{34} +(\)\(81\!\cdots\!00\)\( - 6379590579370173900 \beta) q^{35} +(\)\(30\!\cdots\!08\)\( + 34475654931114098688 \beta) q^{36} +(-\)\(64\!\cdots\!78\)\( + 64367517886888263012 \beta) q^{37} +(\)\(41\!\cdots\!00\)\( - \)\(14\!\cdots\!08\)\( \beta) q^{38} +(-\)\(55\!\cdots\!44\)\( - 99316145693411285198 \beta) q^{39} +(\)\(37\!\cdots\!00\)\( - 48638875975601356800 \beta) q^{40} +(-\)\(55\!\cdots\!78\)\( + \)\(43\!\cdots\!24\)\( \beta) q^{41} +(\)\(12\!\cdots\!28\)\( + \)\(81\!\cdots\!64\)\( \beta) q^{42} +(\)\(52\!\cdots\!76\)\( - \)\(14\!\cdots\!03\)\( \beta) q^{43} +(\)\(88\!\cdots\!72\)\( - \)\(48\!\cdots\!32\)\( \beta) q^{44} +(-\)\(20\!\cdots\!50\)\( - \)\(13\!\cdots\!00\)\( \beta) q^{45} +(\)\(30\!\cdots\!24\)\( - \)\(18\!\cdots\!44\)\( \beta) q^{46} +(\)\(80\!\cdots\!52\)\( + \)\(38\!\cdots\!88\)\( \beta) q^{47} +(-\)\(11\!\cdots\!84\)\( - \)\(47\!\cdots\!96\)\( \beta) q^{48} +(\)\(78\!\cdots\!77\)\( + \)\(18\!\cdots\!92\)\( \beta) q^{49} +(-\)\(16\!\cdots\!00\)\( - \)\(29\!\cdots\!00\)\( \beta) q^{50} +(-\)\(32\!\cdots\!28\)\( - \)\(86\!\cdots\!10\)\( \beta) q^{51} +(-\)\(51\!\cdots\!04\)\( + \)\(47\!\cdots\!08\)\( \beta) q^{52} +(\)\(18\!\cdots\!86\)\( + \)\(11\!\cdots\!88\)\( \beta) q^{53} +(-\)\(10\!\cdots\!40\)\( - \)\(29\!\cdots\!68\)\( \beta) q^{54} +(\)\(42\!\cdots\!00\)\( - \)\(49\!\cdots\!50\)\( \beta) q^{55} +(-\)\(31\!\cdots\!52\)\( - \)\(95\!\cdots\!88\)\( \beta) q^{56} +(\)\(41\!\cdots\!00\)\( - \)\(18\!\cdots\!72\)\( \beta) q^{57} +(-\)\(20\!\cdots\!40\)\( + \)\(26\!\cdots\!64\)\( \beta) q^{58} +(\)\(63\!\cdots\!80\)\( + \)\(85\!\cdots\!57\)\( \beta) q^{59} +(\)\(11\!\cdots\!00\)\( - \)\(97\!\cdots\!00\)\( \beta) q^{60} +(-\)\(65\!\cdots\!58\)\( + \)\(67\!\cdots\!52\)\( \beta) q^{61} +(\)\(63\!\cdots\!68\)\( + \)\(92\!\cdots\!84\)\( \beta) q^{62} +(-\)\(29\!\cdots\!84\)\( - \)\(32\!\cdots\!70\)\( \beta) q^{63} +\)\(32\!\cdots\!56\)\( q^{64} +(-\)\(17\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( \beta) q^{65} +(\)\(69\!\cdots\!48\)\( - \)\(28\!\cdots\!76\)\( \beta) q^{66} +(\)\(19\!\cdots\!52\)\( + \)\(92\!\cdots\!59\)\( \beta) q^{67} +(\)\(56\!\cdots\!52\)\( + \)\(96\!\cdots\!52\)\( \beta) q^{68} +(-\)\(23\!\cdots\!84\)\( - \)\(11\!\cdots\!92\)\( \beta) q^{69} +(\)\(21\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( \beta) q^{70} +(\)\(71\!\cdots\!12\)\( + \)\(13\!\cdots\!62\)\( \beta) q^{71} +(\)\(78\!\cdots\!52\)\( + \)\(90\!\cdots\!72\)\( \beta) q^{72} +(\)\(29\!\cdots\!86\)\( - \)\(21\!\cdots\!32\)\( \beta) q^{73} +(-\)\(16\!\cdots\!32\)\( + \)\(16\!\cdots\!28\)\( \beta) q^{74} +(\)\(24\!\cdots\!00\)\( + \)\(65\!\cdots\!25\)\( \beta) q^{75} +(\)\(10\!\cdots\!00\)\( - \)\(38\!\cdots\!52\)\( \beta) q^{76} +(\)\(86\!\cdots\!44\)\( - \)\(55\!\cdots\!28\)\( \beta) q^{77} +(-\)\(14\!\cdots\!36\)\( - \)\(26\!\cdots\!12\)\( \beta) q^{78} +(-\)\(84\!\cdots\!40\)\( - \)\(15\!\cdots\!32\)\( \beta) q^{79} +(\)\(98\!\cdots\!00\)\( - \)\(12\!\cdots\!00\)\( \beta) q^{80} +(-\)\(12\!\cdots\!99\)\( + \)\(21\!\cdots\!44\)\( \beta) q^{81} +(-\)\(14\!\cdots\!32\)\( + \)\(11\!\cdots\!56\)\( \beta) q^{82} +(\)\(32\!\cdots\!56\)\( - \)\(30\!\cdots\!65\)\( \beta) q^{83} +(\)\(33\!\cdots\!32\)\( + \)\(21\!\cdots\!16\)\( \beta) q^{84} +(\)\(14\!\cdots\!00\)\( - \)\(19\!\cdots\!00\)\( \beta) q^{85} +(\)\(13\!\cdots\!44\)\( - \)\(37\!\cdots\!32\)\( \beta) q^{86} +(-\)\(62\!\cdots\!60\)\( + \)\(52\!\cdots\!86\)\( \beta) q^{87} +(\)\(23\!\cdots\!68\)\( - \)\(12\!\cdots\!08\)\( \beta) q^{88} +(\)\(34\!\cdots\!90\)\( + \)\(15\!\cdots\!40\)\( \beta) q^{89} +(-\)\(53\!\cdots\!00\)\( - \)\(34\!\cdots\!00\)\( \beta) q^{90} +(-\)\(29\!\cdots\!08\)\( - \)\(82\!\cdots\!36\)\( \beta) q^{91} +(\)\(80\!\cdots\!56\)\( - \)\(49\!\cdots\!36\)\( \beta) q^{92} +(-\)\(29\!\cdots\!88\)\( - \)\(11\!\cdots\!16\)\( \beta) q^{93} +(\)\(21\!\cdots\!88\)\( + \)\(10\!\cdots\!72\)\( \beta) q^{94} +(\)\(15\!\cdots\!00\)\( - \)\(15\!\cdots\!50\)\( \beta) q^{95} +(-\)\(31\!\cdots\!96\)\( - \)\(12\!\cdots\!24\)\( \beta) q^{96} +(\)\(29\!\cdots\!22\)\( + \)\(20\!\cdots\!80\)\( \beta) q^{97} +(\)\(20\!\cdots\!88\)\( + \)\(48\!\cdots\!48\)\( \beta) q^{98} +(\)\(26\!\cdots\!56\)\( + \)\(33\!\cdots\!05\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 524288q^{2} - 501686808q^{3} + 137438953472q^{4} + 4182600859500q^{5} - 131514186596352q^{6} - 3512678957537456q^{7} + 36028797018963968q^{8} + 873634568291944506q^{9} + O(q^{10}) \) \( 2q + 524288q^{2} - 501686808q^{3} + 137438953472q^{4} + 4182600859500q^{5} - 131514186596352q^{6} - 3512678957537456q^{7} + 36028797018963968q^{8} + 873634568291944506q^{9} + 1096443719712768000q^{10} + 25659945722560373304q^{11} - 34475654931114098688q^{12} - \)\(15\!\cdots\!28\)\(q^{13} - \)\(92\!\cdots\!64\)\(q^{14} + \)\(34\!\cdots\!00\)\(q^{15} + \)\(94\!\cdots\!92\)\(q^{16} + \)\(16\!\cdots\!64\)\(q^{17} + \)\(22\!\cdots\!64\)\(q^{18} + \)\(31\!\cdots\!00\)\(q^{19} + \)\(28\!\cdots\!00\)\(q^{20} + \)\(96\!\cdots\!24\)\(q^{21} + \)\(67\!\cdots\!76\)\(q^{22} + \)\(23\!\cdots\!92\)\(q^{23} - \)\(90\!\cdots\!72\)\(q^{24} - \)\(12\!\cdots\!50\)\(q^{25} - \)\(39\!\cdots\!32\)\(q^{26} - \)\(82\!\cdots\!20\)\(q^{27} - \)\(24\!\cdots\!16\)\(q^{28} - \)\(15\!\cdots\!20\)\(q^{29} + \)\(89\!\cdots\!00\)\(q^{30} + \)\(48\!\cdots\!44\)\(q^{31} + \)\(24\!\cdots\!48\)\(q^{32} + \)\(52\!\cdots\!84\)\(q^{33} + \)\(43\!\cdots\!16\)\(q^{34} + \)\(16\!\cdots\!00\)\(q^{35} + \)\(60\!\cdots\!16\)\(q^{36} - \)\(12\!\cdots\!56\)\(q^{37} + \)\(82\!\cdots\!00\)\(q^{38} - \)\(11\!\cdots\!88\)\(q^{39} + \)\(75\!\cdots\!00\)\(q^{40} - \)\(11\!\cdots\!56\)\(q^{41} + \)\(25\!\cdots\!56\)\(q^{42} + \)\(10\!\cdots\!52\)\(q^{43} + \)\(17\!\cdots\!44\)\(q^{44} - \)\(40\!\cdots\!00\)\(q^{45} + \)\(61\!\cdots\!48\)\(q^{46} + \)\(16\!\cdots\!04\)\(q^{47} - \)\(23\!\cdots\!68\)\(q^{48} + \)\(15\!\cdots\!54\)\(q^{49} - \)\(32\!\cdots\!00\)\(q^{50} - \)\(64\!\cdots\!56\)\(q^{51} - \)\(10\!\cdots\!08\)\(q^{52} + \)\(37\!\cdots\!72\)\(q^{53} - \)\(21\!\cdots\!80\)\(q^{54} + \)\(85\!\cdots\!00\)\(q^{55} - \)\(63\!\cdots\!04\)\(q^{56} + \)\(83\!\cdots\!00\)\(q^{57} - \)\(40\!\cdots\!80\)\(q^{58} + \)\(12\!\cdots\!60\)\(q^{59} + \)\(23\!\cdots\!00\)\(q^{60} - \)\(13\!\cdots\!16\)\(q^{61} + \)\(12\!\cdots\!36\)\(q^{62} - \)\(59\!\cdots\!68\)\(q^{63} + \)\(64\!\cdots\!12\)\(q^{64} - \)\(34\!\cdots\!00\)\(q^{65} + \)\(13\!\cdots\!96\)\(q^{66} + \)\(38\!\cdots\!04\)\(q^{67} + \)\(11\!\cdots\!04\)\(q^{68} - \)\(46\!\cdots\!68\)\(q^{69} + \)\(42\!\cdots\!00\)\(q^{70} + \)\(14\!\cdots\!24\)\(q^{71} + \)\(15\!\cdots\!04\)\(q^{72} + \)\(58\!\cdots\!72\)\(q^{73} - \)\(33\!\cdots\!64\)\(q^{74} + \)\(49\!\cdots\!00\)\(q^{75} + \)\(21\!\cdots\!00\)\(q^{76} + \)\(17\!\cdots\!88\)\(q^{77} - \)\(29\!\cdots\!72\)\(q^{78} - \)\(16\!\cdots\!80\)\(q^{79} + \)\(19\!\cdots\!00\)\(q^{80} - \)\(24\!\cdots\!98\)\(q^{81} - \)\(29\!\cdots\!64\)\(q^{82} + \)\(65\!\cdots\!12\)\(q^{83} + \)\(66\!\cdots\!64\)\(q^{84} + \)\(28\!\cdots\!00\)\(q^{85} + \)\(27\!\cdots\!88\)\(q^{86} - \)\(12\!\cdots\!20\)\(q^{87} + \)\(46\!\cdots\!36\)\(q^{88} + \)\(69\!\cdots\!80\)\(q^{89} - \)\(10\!\cdots\!00\)\(q^{90} - \)\(58\!\cdots\!16\)\(q^{91} + \)\(16\!\cdots\!12\)\(q^{92} - \)\(59\!\cdots\!76\)\(q^{93} + \)\(42\!\cdots\!76\)\(q^{94} + \)\(31\!\cdots\!00\)\(q^{95} - \)\(62\!\cdots\!92\)\(q^{96} + \)\(58\!\cdots\!44\)\(q^{97} + \)\(41\!\cdots\!76\)\(q^{98} + \)\(53\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

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
236418.
−236417.
262144. −1.15869e9 6.87195e10 −3.59875e11 −3.03743e14 −6.58432e15 1.80144e16 8.92270e17 −9.43392e16
1.2 262144. 6.57000e8 6.87195e10 4.54248e12 1.72228e14 3.07164e15 1.80144e16 −1.86355e16 1.19078e18
\(n\): 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 \( T_{3}^{2} + 501686808 T_{3} - \)\(76\!\cdots\!84\)\( \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(2))\).