Properties

Label 2.38.a.a
Level 2
Weight 38
Character orbit 2.a
Self dual Yes
Analytic conductor 17.343
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 5 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 17280\sqrt{3026574721}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -262144 q^{2} \) \( + ( 211535604 - \beta ) q^{3} \) \( + 68719476736 q^{4} \) \( + ( -6753765277770 + 3444 \beta ) q^{5} \) \( + ( -55452789374976 + 262144 \beta ) q^{6} \) \( + ( 1553087081481128 + 5126718 \beta ) q^{7} \) \( -18014398509481984 q^{8} \) \( + ( 498193775039693853 - 423071208 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(-262144 q^{2}\) \(+(211535604 - \beta) q^{3}\) \(+68719476736 q^{4}\) \(+(-6753765277770 + 3444 \beta) q^{5}\) \(+(-55452789374976 + 262144 \beta) q^{6}\) \(+(1553087081481128 + 5126718 \beta) q^{7}\) \(-18014398509481984 q^{8}\) \(+(498193775039693853 - 423071208 \beta) q^{9}\) \(+(1770459044975738880 - 902823936 \beta) q^{10}\) \(+(2001439330770516252 + 25266324237 \beta) q^{11}\) \(+(14536616017913708544 - 68719476736 \beta) q^{12}\) \(+(-\)\(20\!\cdots\!86\)\( + 40178550228 \beta) q^{13}\) \(+(-\)\(40\!\cdots\!32\)\( - 1343938363392 \beta) q^{14}\) \(+(-\)\(45\!\cdots\!80\)\( + 7482293897946 \beta) q^{15}\) \(+\)\(47\!\cdots\!96\)\( q^{16}\) \(+(\)\(46\!\cdots\!18\)\( - 73067502067368 \beta) q^{17}\) \(+(-\)\(13\!\cdots\!32\)\( + 110905578749952 \beta) q^{18}\) \(+(-\)\(75\!\cdots\!00\)\( + 83542782562107 \beta) q^{19}\) \(+(-\)\(46\!\cdots\!20\)\( + 236669877878784 \beta) q^{20}\) \(+(-\)\(43\!\cdots\!88\)\( - 468603692813456 \beta) q^{21}\) \(+(-\)\(52\!\cdots\!88\)\( - 6623415300784128 \beta) q^{22}\) \(+(\)\(24\!\cdots\!04\)\( + 13722480149762874 \beta) q^{23}\) \(+(-\)\(38\!\cdots\!36\)\( + 18014398509481984 \beta) q^{24}\) \(+(-\)\(16\!\cdots\!25\)\( - 46519935233279760 \beta) q^{25}\) \(+(\)\(54\!\cdots\!84\)\( - 10532565870968832 \beta) q^{26}\) \(+(\)\(39\!\cdots\!60\)\( - 137404492667986122 \beta) q^{27}\) \(+(\)\(10\!\cdots\!08\)\( + 352305378333032448 \beta) q^{28}\) \(+(\)\(58\!\cdots\!90\)\( + 872730741021530244 \beta) q^{29}\) \(+(\)\(11\!\cdots\!20\)\( - 1961438451583156224 \beta) q^{30}\) \(+(-\)\(27\!\cdots\!28\)\( - 872718796832206536 \beta) q^{31}\) \(-\)\(12\!\cdots\!24\)\( q^{32}\) \(+(-\)\(22\!\cdots\!92\)\( + 3343287827563117896 \beta) q^{33}\) \(+(-\)\(12\!\cdots\!92\)\( + 19154207261948116992 \beta) q^{34}\) \(+(\)\(54\!\cdots\!40\)\( - 29275818108697454028 \beta) q^{35}\) \(+(\)\(34\!\cdots\!08\)\( - 29073232035827417088 \beta) q^{36}\) \(+(-\)\(57\!\cdots\!22\)\( + 28974975230710445412 \beta) q^{37}\) \(+(\)\(19\!\cdots\!00\)\( - 21900239191960977408 \beta) q^{38}\) \(+(-\)\(80\!\cdots\!44\)\( + \)\(21\!\cdots\!98\)\( \beta) q^{39}\) \(+(\)\(12\!\cdots\!80\)\( - 62041588466655952896 \beta) q^{40}\) \(+(\)\(23\!\cdots\!22\)\( - \)\(33\!\cdots\!24\)\( \beta) q^{41}\) \(+(\)\(11\!\cdots\!72\)\( + \)\(12\!\cdots\!64\)\( \beta) q^{42}\) \(+(-\)\(15\!\cdots\!76\)\( - \)\(13\!\cdots\!03\)\( \beta) q^{43}\) \(+(\)\(13\!\cdots\!72\)\( + \)\(17\!\cdots\!32\)\( \beta) q^{44}\) \(+(-\)\(46\!\cdots\!10\)\( + \)\(45\!\cdots\!92\)\( \beta) q^{45}\) \(+(-\)\(65\!\cdots\!76\)\( - \)\(35\!\cdots\!56\)\( \beta) q^{46}\) \(+(-\)\(56\!\cdots\!52\)\( - \)\(75\!\cdots\!12\)\( \beta) q^{47}\) \(+(\)\(99\!\cdots\!84\)\( - \)\(47\!\cdots\!96\)\( \beta) q^{48}\) \(+(\)\(76\!\cdots\!77\)\( + \)\(15\!\cdots\!08\)\( \beta) q^{49}\) \(+(\)\(43\!\cdots\!00\)\( + \)\(12\!\cdots\!40\)\( \beta) q^{50}\) \(+(\)\(66\!\cdots\!72\)\( - \)\(15\!\cdots\!90\)\( \beta) q^{51}\) \(+(-\)\(14\!\cdots\!96\)\( + \)\(27\!\cdots\!08\)\( \beta) q^{52}\) \(+(\)\(26\!\cdots\!14\)\( - \)\(10\!\cdots\!12\)\( \beta) q^{53}\) \(+(-\)\(10\!\cdots\!40\)\( + \)\(36\!\cdots\!68\)\( \beta) q^{54}\) \(+(\)\(65\!\cdots\!60\)\( - \)\(16\!\cdots\!02\)\( \beta) q^{55}\) \(+(-\)\(27\!\cdots\!52\)\( - \)\(92\!\cdots\!12\)\( \beta) q^{56}\) \(+(-\)\(23\!\cdots\!00\)\( + \)\(76\!\cdots\!28\)\( \beta) q^{57}\) \(+(-\)\(15\!\cdots\!60\)\( - \)\(22\!\cdots\!36\)\( \beta) q^{58}\) \(+(\)\(29\!\cdots\!80\)\( - \)\(81\!\cdots\!07\)\( \beta) q^{59}\) \(+(-\)\(31\!\cdots\!80\)\( + \)\(51\!\cdots\!56\)\( \beta) q^{60}\) \(+(\)\(47\!\cdots\!42\)\( - \)\(77\!\cdots\!52\)\( \beta) q^{61}\) \(+(\)\(72\!\cdots\!32\)\( + \)\(22\!\cdots\!84\)\( \beta) q^{62}\) \(+(-\)\(11\!\cdots\!16\)\( + \)\(18\!\cdots\!30\)\( \beta) q^{63}\) \(+\)\(32\!\cdots\!56\)\( q^{64}\) \(+(\)\(15\!\cdots\!20\)\( - \)\(99\!\cdots\!44\)\( \beta) q^{65}\) \(+(\)\(58\!\cdots\!48\)\( - \)\(87\!\cdots\!24\)\( \beta) q^{66}\) \(+(\)\(16\!\cdots\!48\)\( + \)\(19\!\cdots\!59\)\( \beta) q^{67}\) \(+(\)\(31\!\cdots\!48\)\( - \)\(50\!\cdots\!48\)\( \beta) q^{68}\) \(+(-\)\(11\!\cdots\!84\)\( + \)\(41\!\cdots\!92\)\( \beta) q^{69}\) \(+(-\)\(14\!\cdots\!60\)\( + \)\(76\!\cdots\!32\)\( \beta) q^{70}\) \(+(-\)\(18\!\cdots\!88\)\( + \)\(11\!\cdots\!38\)\( \beta) q^{71}\) \(+(-\)\(89\!\cdots\!52\)\( + \)\(76\!\cdots\!72\)\( \beta) q^{72}\) \(+(-\)\(53\!\cdots\!86\)\( - \)\(38\!\cdots\!32\)\( \beta) q^{73}\) \(+(\)\(15\!\cdots\!68\)\( - \)\(75\!\cdots\!28\)\( \beta) q^{74}\) \(+(\)\(38\!\cdots\!00\)\( + \)\(65\!\cdots\!85\)\( \beta) q^{75}\) \(+(-\)\(51\!\cdots\!00\)\( + \)\(57\!\cdots\!52\)\( \beta) q^{76}\) \(+(\)\(12\!\cdots\!56\)\( + \)\(49\!\cdots\!72\)\( \beta) q^{77}\) \(+(\)\(21\!\cdots\!36\)\( - \)\(56\!\cdots\!12\)\( \beta) q^{78}\) \(+(\)\(20\!\cdots\!60\)\( + \)\(10\!\cdots\!32\)\( \beta) q^{79}\) \(+(-\)\(31\!\cdots\!20\)\( + \)\(16\!\cdots\!24\)\( \beta) q^{80}\) \(+(-\)\(17\!\cdots\!99\)\( - \)\(23\!\cdots\!44\)\( \beta) q^{81}\) \(+(-\)\(62\!\cdots\!68\)\( + \)\(88\!\cdots\!56\)\( \beta) q^{82}\) \(+(\)\(86\!\cdots\!44\)\( - \)\(17\!\cdots\!65\)\( \beta) q^{83}\) \(+(-\)\(29\!\cdots\!68\)\( - \)\(32\!\cdots\!16\)\( \beta) q^{84}\) \(+(-\)\(23\!\cdots\!60\)\( + \)\(49\!\cdots\!52\)\( \beta) q^{85}\) \(+(\)\(41\!\cdots\!44\)\( + \)\(35\!\cdots\!32\)\( \beta) q^{86}\) \(+(-\)\(66\!\cdots\!40\)\( - \)\(40\!\cdots\!14\)\( \beta) q^{87}\) \(+(-\)\(36\!\cdots\!68\)\( - \)\(45\!\cdots\!08\)\( \beta) q^{88}\) \(+(-\)\(38\!\cdots\!10\)\( + \)\(65\!\cdots\!60\)\( \beta) q^{89}\) \(+(\)\(12\!\cdots\!40\)\( - \)\(11\!\cdots\!48\)\( \beta) q^{90}\) \(+(-\)\(13\!\cdots\!08\)\( - \)\(10\!\cdots\!64\)\( \beta) q^{91}\) \(+(\)\(17\!\cdots\!44\)\( + \)\(94\!\cdots\!64\)\( \beta) q^{92}\) \(+(\)\(20\!\cdots\!88\)\( + \)\(25\!\cdots\!84\)\( \beta) q^{93}\) \(+(\)\(14\!\cdots\!88\)\( + \)\(19\!\cdots\!28\)\( \beta) q^{94}\) \(+(\)\(53\!\cdots\!00\)\( - \)\(31\!\cdots\!90\)\( \beta) q^{95}\) \(+(-\)\(26\!\cdots\!96\)\( + \)\(12\!\cdots\!24\)\( \beta) q^{96}\) \(+(-\)\(32\!\cdots\!22\)\( - \)\(57\!\cdots\!20\)\( \beta) q^{97}\) \(+(-\)\(19\!\cdots\!88\)\( - \)\(41\!\cdots\!52\)\( \beta) q^{98}\) \(+(-\)\(86\!\cdots\!44\)\( + \)\(11\!\cdots\!45\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 524288q^{2} \) \(\mathstrut +\mathstrut 423071208q^{3} \) \(\mathstrut +\mathstrut 137438953472q^{4} \) \(\mathstrut -\mathstrut 13507530555540q^{5} \) \(\mathstrut -\mathstrut 110905578749952q^{6} \) \(\mathstrut +\mathstrut 3106174162962256q^{7} \) \(\mathstrut -\mathstrut 36028797018963968q^{8} \) \(\mathstrut +\mathstrut 996387550079387706q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 524288q^{2} \) \(\mathstrut +\mathstrut 423071208q^{3} \) \(\mathstrut +\mathstrut 137438953472q^{4} \) \(\mathstrut -\mathstrut 13507530555540q^{5} \) \(\mathstrut -\mathstrut 110905578749952q^{6} \) \(\mathstrut +\mathstrut 3106174162962256q^{7} \) \(\mathstrut -\mathstrut 36028797018963968q^{8} \) \(\mathstrut +\mathstrut 996387550079387706q^{9} \) \(\mathstrut +\mathstrut 3540918089951477760q^{10} \) \(\mathstrut +\mathstrut 4002878661541032504q^{11} \) \(\mathstrut +\mathstrut 29073232035827417088q^{12} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!72\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!64\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!60\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!92\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!36\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!64\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!40\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!76\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!76\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!08\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!72\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!68\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!20\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!16\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!40\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!56\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!48\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!84\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!84\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!16\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!44\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!00\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!88\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!60\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!44\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!44\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!52\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!44\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!20\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!52\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!04\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!68\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!54\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!44\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!92\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!28\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!80\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!20\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!04\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!20\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!60\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!60\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!84\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!64\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!32\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!12\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!40\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!96\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!96\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!96\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!68\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!20\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!76\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!04\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!72\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!36\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!12\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!72\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!20\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!40\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!98\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!36\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!88\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!36\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!20\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!88\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!80\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!36\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!20\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!80\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!16\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!88\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!76\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!76\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!92\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!44\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!76\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
27507.7
−27506.7
−262144. −7.39112e8 6.87195e10 −3.47974e12 1.93754e14 6.42679e15 −1.80144e16 9.60023e16 9.12192e17
1.2 −262144. 1.16218e9 6.87195e10 −1.00278e13 −3.04659e14 −3.32061e15 −1.80144e16 9.00385e17 2.62873e18
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 423071208 T_{3} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!84\)\( \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(2))\).