Properties

Label 2.34.a.b
Level 2
Weight 34
Character orbit 2.a
Self dual Yes
Analytic conductor 13.797
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 34 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(13.7965657762\)
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: \( 2^{7}\cdot 3\cdot 5\cdot 11 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 10560\sqrt{79829689}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 65536 q^{2} \) \( + ( 4178244 - \beta ) q^{3} \) \( + 4294967296 q^{4} \) \( + ( -2666238330 - 3996 \beta ) q^{5} \) \( + ( 273825398784 - 65536 \beta ) q^{6} \) \( + ( 66359547937928 + 896238 \beta ) q^{7} \) \( + 281474976710656 q^{8} \) \( + ( 3360493163638413 - 8356488 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+65536 q^{2}\) \(+(4178244 - \beta) q^{3}\) \(+4294967296 q^{4}\) \(+(-2666238330 - 3996 \beta) q^{5}\) \(+(273825398784 - 65536 \beta) q^{6}\) \(+(66359547937928 + 896238 \beta) q^{7}\) \(+281474976710656 q^{8}\) \(+(3360493163638413 - 8356488 \beta) q^{9}\) \(+(-174734595194880 - 261881856 \beta) q^{10}\) \(+(-79052735711361588 + 2213453277 \beta) q^{11}\) \(+(17945421334708224 - 4294967296 \beta) q^{12}\) \(+(2491699257749546894 - 13597194492 \beta) q^{13}\) \(+(4348939333660049408 + 58735853568 \beta) q^{14}\) \(+(35561635450747625880 - 14030024694 \beta) q^{15}\) \(+18446744073709551616 q^{16}\) \(+(\)\(15\!\cdots\!38\)\( - 833438065608 \beta) q^{17}\) \(+(\)\(22\!\cdots\!68\)\( - 547650797568 \beta) q^{18}\) \(+(\)\(11\!\cdots\!00\)\( + 13790479962267 \beta) q^{19}\) \(+(-11451406430691655680 - 17162689314816 \beta) q^{20}\) \(+(-\)\(77\!\cdots\!68\)\( - 62614846891856 \beta) q^{21}\) \(+(-\)\(51\!\cdots\!68\)\( + 145060873961472 \beta) q^{22}\) \(+(-\)\(35\!\cdots\!76\)\( + 63128735356554 \beta) q^{23}\) \(+(\)\(11\!\cdots\!64\)\( - 281474976710656 \beta) q^{24}\) \(+(\)\(25\!\cdots\!75\)\( + 21308576733360 \beta) q^{25}\) \(+(\)\(16\!\cdots\!84\)\( - 891105738227712 \beta) q^{26}\) \(+(\)\(65\!\cdots\!60\)\( + 2163651957070038 \beta) q^{27}\) \(+(\)\(28\!\cdots\!88\)\( + 3849312899432448 \beta) q^{28}\) \(+(-\)\(12\!\cdots\!10\)\( - 10405860886172556 \beta) q^{29}\) \(+(\)\(23\!\cdots\!80\)\( - 919471698345984 \beta) q^{30}\) \(+(-\)\(21\!\cdots\!48\)\( + 2188390162942584 \beta) q^{31}\) \(+\)\(12\!\cdots\!76\)\( q^{32}\) \(+(-\)\(20\!\cdots\!72\)\( + 88301083585267176 \beta) q^{33}\) \(+(\)\(99\!\cdots\!68\)\( - 54620197067685888 \beta) q^{34}\) \(+(-\)\(32\!\cdots\!40\)\( - 267562337668362828 \beta) q^{35}\) \(+(\)\(14\!\cdots\!48\)\( - 35890842669416448 \beta) q^{36}\) \(+(\)\(58\!\cdots\!38\)\( + 704679249787856532 \beta) q^{37}\) \(+(\)\(73\!\cdots\!00\)\( + 903772894807130112 \beta) q^{38}\) \(+(\)\(13\!\cdots\!36\)\( - 2548511654052578942 \beta) q^{39}\) \(+(-\)\(75\!\cdots\!80\)\( - 1124774006935781376 \beta) q^{40}\) \(+(-\)\(83\!\cdots\!38\)\( + 5060866057679897136 \beta) q^{41}\) \(+(-\)\(50\!\cdots\!48\)\( - 4103526605904674816 \beta) q^{42}\) \(+(-\)\(18\!\cdots\!76\)\( + 795637191126404277 \beta) q^{43}\) \(+(-\)\(33\!\cdots\!48\)\( + 9506709435939028992 \beta) q^{44}\) \(+(\)\(28\!\cdots\!10\)\( - 13406250293289313308 \beta) q^{45}\) \(+(-\)\(23\!\cdots\!36\)\( + 4137204800327122944 \beta) q^{46}\) \(+(\)\(14\!\cdots\!28\)\( - 18059171324272980012 \beta) q^{47}\) \(+(\)\(77\!\cdots\!04\)\( - 18446744073709551616 \beta) q^{48}\) \(+(\)\(38\!\cdots\!77\)\( + \)\(11\!\cdots\!28\)\( \beta) q^{49}\) \(+(\)\(16\!\cdots\!00\)\( + 1396478884797480960 \beta) q^{50}\) \(+(\)\(80\!\cdots\!72\)\( - \)\(15\!\cdots\!90\)\( \beta) q^{51}\) \(+(\)\(10\!\cdots\!24\)\( - 58399505660491333632 \beta) q^{52}\) \(+(-\)\(21\!\cdots\!46\)\( + 12899519303831752788 \beta) q^{53}\) \(+(\)\(42\!\cdots\!60\)\( + \)\(14\!\cdots\!68\)\( \beta) q^{54}\) \(+(-\)\(78\!\cdots\!60\)\( + \)\(30\!\cdots\!38\)\( \beta) q^{55}\) \(+(\)\(18\!\cdots\!68\)\( + \)\(25\!\cdots\!28\)\( \beta) q^{56}\) \(+(-\)\(11\!\cdots\!00\)\( - \)\(10\!\cdots\!52\)\( \beta) q^{57}\) \(+(-\)\(82\!\cdots\!60\)\( - \)\(68\!\cdots\!16\)\( \beta) q^{58}\) \(+(\)\(93\!\cdots\!80\)\( + \)\(15\!\cdots\!73\)\( \beta) q^{59}\) \(+(\)\(15\!\cdots\!80\)\( - 60258497222802407424 \beta) q^{60}\) \(+(\)\(23\!\cdots\!02\)\( - \)\(42\!\cdots\!92\)\( \beta) q^{61}\) \(+(-\)\(14\!\cdots\!28\)\( + \)\(14\!\cdots\!24\)\( \beta) q^{62}\) \(+(\)\(15\!\cdots\!64\)\( + \)\(24\!\cdots\!30\)\( \beta) q^{63}\) \(+\)\(79\!\cdots\!36\)\( q^{64}\) \(+(\)\(47\!\cdots\!80\)\( - \)\(99\!\cdots\!64\)\( \beta) q^{65}\) \(+(-\)\(13\!\cdots\!92\)\( + \)\(57\!\cdots\!36\)\( \beta) q^{66}\) \(+(-\)\(37\!\cdots\!32\)\( - \)\(98\!\cdots\!21\)\( \beta) q^{67}\) \(+(\)\(64\!\cdots\!48\)\( - \)\(35\!\cdots\!68\)\( \beta) q^{68}\) \(+(-\)\(71\!\cdots\!44\)\( + \)\(35\!\cdots\!52\)\( \beta) q^{69}\) \(+(-\)\(21\!\cdots\!40\)\( - \)\(17\!\cdots\!08\)\( \beta) q^{70}\) \(+(\)\(13\!\cdots\!32\)\( - \)\(32\!\cdots\!62\)\( \beta) q^{71}\) \(+(\)\(94\!\cdots\!28\)\( - \)\(23\!\cdots\!28\)\( \beta) q^{72}\) \(+(-\)\(26\!\cdots\!66\)\( - \)\(70\!\cdots\!72\)\( \beta) q^{73}\) \(+(\)\(38\!\cdots\!68\)\( + \)\(46\!\cdots\!52\)\( \beta) q^{74}\) \(+(-\)\(82\!\cdots\!00\)\( - \)\(25\!\cdots\!35\)\( \beta) q^{75}\) \(+(\)\(48\!\cdots\!00\)\( + \)\(59\!\cdots\!32\)\( \beta) q^{76}\) \(+(\)\(12\!\cdots\!36\)\( + \)\(76\!\cdots\!12\)\( \beta) q^{77}\) \(+(\)\(86\!\cdots\!96\)\( - \)\(16\!\cdots\!12\)\( \beta) q^{78}\) \(+(-\)\(13\!\cdots\!40\)\( - \)\(55\!\cdots\!68\)\( \beta) q^{79}\) \(+(-\)\(49\!\cdots\!80\)\( - \)\(73\!\cdots\!36\)\( \beta) q^{80}\) \(+(-\)\(37\!\cdots\!59\)\( - \)\(97\!\cdots\!64\)\( \beta) q^{81}\) \(+(-\)\(54\!\cdots\!68\)\( + \)\(33\!\cdots\!96\)\( \beta) q^{82}\) \(+(-\)\(18\!\cdots\!96\)\( + \)\(35\!\cdots\!35\)\( \beta) q^{83}\) \(+(-\)\(33\!\cdots\!28\)\( - \)\(26\!\cdots\!76\)\( \beta) q^{84}\) \(+(\)\(29\!\cdots\!60\)\( - \)\(60\!\cdots\!08\)\( \beta) q^{85}\) \(+(-\)\(12\!\cdots\!36\)\( + \)\(52\!\cdots\!72\)\( \beta) q^{86}\) \(+(\)\(92\!\cdots\!60\)\( + \)\(82\!\cdots\!46\)\( \beta) q^{87}\) \(+(-\)\(22\!\cdots\!28\)\( + \)\(62\!\cdots\!12\)\( \beta) q^{88}\) \(+(\)\(15\!\cdots\!90\)\( - \)\(50\!\cdots\!40\)\( \beta) q^{89}\) \(+(\)\(18\!\cdots\!60\)\( - \)\(87\!\cdots\!88\)\( \beta) q^{90}\) \(+(\)\(56\!\cdots\!32\)\( + \)\(13\!\cdots\!96\)\( \beta) q^{91}\) \(+(-\)\(15\!\cdots\!96\)\( + \)\(27\!\cdots\!84\)\( \beta) q^{92}\) \(+(-\)\(28\!\cdots\!12\)\( + \)\(21\!\cdots\!44\)\( \beta) q^{93}\) \(+(\)\(94\!\cdots\!08\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{94}\) \(+(-\)\(49\!\cdots\!00\)\( - \)\(45\!\cdots\!10\)\( \beta) q^{95}\) \(+(\)\(50\!\cdots\!44\)\( - \)\(12\!\cdots\!76\)\( \beta) q^{96}\) \(+(\)\(34\!\cdots\!58\)\( - \)\(24\!\cdots\!20\)\( \beta) q^{97}\) \(+(\)\(25\!\cdots\!72\)\( + \)\(77\!\cdots\!08\)\( \beta) q^{98}\) \(+(-\)\(43\!\cdots\!44\)\( + \)\(80\!\cdots\!45\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 131072q^{2} \) \(\mathstrut +\mathstrut 8356488q^{3} \) \(\mathstrut +\mathstrut 8589934592q^{4} \) \(\mathstrut -\mathstrut 5332476660q^{5} \) \(\mathstrut +\mathstrut 547650797568q^{6} \) \(\mathstrut +\mathstrut 132719095875856q^{7} \) \(\mathstrut +\mathstrut 562949953421312q^{8} \) \(\mathstrut +\mathstrut 6720986327276826q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 131072q^{2} \) \(\mathstrut +\mathstrut 8356488q^{3} \) \(\mathstrut +\mathstrut 8589934592q^{4} \) \(\mathstrut -\mathstrut 5332476660q^{5} \) \(\mathstrut +\mathstrut 547650797568q^{6} \) \(\mathstrut +\mathstrut 132719095875856q^{7} \) \(\mathstrut +\mathstrut 562949953421312q^{8} \) \(\mathstrut +\mathstrut 6720986327276826q^{9} \) \(\mathstrut -\mathstrut 349469190389760q^{10} \) \(\mathstrut -\mathstrut 158105471422723176q^{11} \) \(\mathstrut +\mathstrut 35890842669416448q^{12} \) \(\mathstrut +\mathstrut 4983398515499093788q^{13} \) \(\mathstrut +\mathstrut 8697878667320098816q^{14} \) \(\mathstrut +\mathstrut 71123270901495251760q^{15} \) \(\mathstrut +\mathstrut 36893488147419103232q^{16} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!76\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!36\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut 22902812861383311360q^{20} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!36\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!36\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!52\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!28\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!68\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!20\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!76\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!20\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!60\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!96\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!52\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!44\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!36\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!80\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!96\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!76\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!72\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!60\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!76\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!96\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!52\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!96\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!20\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!72\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!56\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!08\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!54\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!44\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!48\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!92\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!20\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!20\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!36\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!20\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!60\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!60\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!04\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!56\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!28\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!72\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!60\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!84\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!64\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!96\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!88\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!80\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!64\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!56\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!32\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!36\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!72\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!92\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!80\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!60\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!18\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!36\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!92\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!56\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!20\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!72\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!20\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!56\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!80\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!20\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!64\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!92\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!24\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!16\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!88\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!16\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!44\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!88\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
4467.87
−4466.87
65536.0 −9.01727e7 4.29497e9 −3.79693e11 −5.90956e12 1.50920e14 2.81475e14 2.57205e15 −2.48835e16
1.2 65536.0 9.85292e7 4.29497e9 3.74360e11 6.45721e12 −1.82013e13 2.81475e14 4.14894e15 2.45341e16
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 8356488 T_{3} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!64\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(2))\).