Properties

Label 2.26.a.a
Level 2
Weight 26
Character orbit 2.a
Self dual Yes
Analytic conductor 7.920
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 26 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(7.91993559904\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 4096q^{2} \) \(\mathstrut +\mathstrut 97956q^{3} \) \(\mathstrut +\mathstrut 16777216q^{4} \) \(\mathstrut +\mathstrut 341005350q^{5} \) \(\mathstrut -\mathstrut 401227776q^{6} \) \(\mathstrut -\mathstrut 40882637368q^{7} \) \(\mathstrut -\mathstrut 68719476736q^{8} \) \(\mathstrut -\mathstrut 837693231507q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 4096q^{2} \) \(\mathstrut +\mathstrut 97956q^{3} \) \(\mathstrut +\mathstrut 16777216q^{4} \) \(\mathstrut +\mathstrut 341005350q^{5} \) \(\mathstrut -\mathstrut 401227776q^{6} \) \(\mathstrut -\mathstrut 40882637368q^{7} \) \(\mathstrut -\mathstrut 68719476736q^{8} \) \(\mathstrut -\mathstrut 837693231507q^{9} \) \(\mathstrut -\mathstrut 1396757913600q^{10} \) \(\mathstrut -\mathstrut 14506222377108q^{11} \) \(\mathstrut +\mathstrut 1643428970496q^{12} \) \(\mathstrut +\mathstrut 87843989537006q^{13} \) \(\mathstrut +\mathstrut 167455282659328q^{14} \) \(\mathstrut +\mathstrut 33403520064600q^{15} \) \(\mathstrut +\mathstrut 281474976710656q^{16} \) \(\mathstrut -\mathstrut 2655425868886638q^{17} \) \(\mathstrut +\mathstrut 3431191476252672q^{18} \) \(\mathstrut -\mathstrut 13999424704097740q^{19} \) \(\mathstrut +\mathstrut 5721120414105600q^{20} \) \(\mathstrut -\mathstrut 4004699626019808q^{21} \) \(\mathstrut +\mathstrut 59417486856634368q^{22} \) \(\mathstrut +\mathstrut 85852780724035416q^{23} \) \(\mathstrut -\mathstrut 6731485063151616q^{24} \) \(\mathstrut -\mathstrut 181738575148330625q^{25} \) \(\mathstrut -\mathstrut 359808981143576576q^{26} \) \(\mathstrut -\mathstrut 165054081212098200q^{27} \) \(\mathstrut -\mathstrut 685896837772607488q^{28} \) \(\mathstrut +\mathstrut 2080230429601526910q^{29} \) \(\mathstrut -\mathstrut 136820818184601600q^{30} \) \(\mathstrut +\mathstrut 2663532371302675232q^{31} \) \(\mathstrut -\mathstrut 1152921504606846976q^{32} \) \(\mathstrut -\mathstrut 1420971519171991248q^{33} \) \(\mathstrut +\mathstrut 10876624358959669248q^{34} \) \(\mathstrut -\mathstrut 13941198064597918800q^{35} \) \(\mathstrut -\mathstrut 14054160286730944512q^{36} \) \(\mathstrut -\mathstrut 51379607980315436218q^{37} \) \(\mathstrut +\mathstrut 57341643587984343040q^{38} \) \(\mathstrut +\mathstrut 8604845839086959736q^{39} \) \(\mathstrut -\mathstrut 23433709216176537600q^{40} \) \(\mathstrut +\mathstrut 233559807284225330922q^{41} \) \(\mathstrut +\mathstrut 16403249668177133568q^{42} \) \(\mathstrut -\mathstrut 40133597094729613684q^{43} \) \(\mathstrut -\mathstrut 243374026164774371328q^{44} \) \(\mathstrut -\mathstrut 285657873602675562450q^{45} \) \(\mathstrut -\mathstrut 351652989845649063936q^{46} \) \(\mathstrut +\mathstrut 279826395037007620272q^{47} \) \(\mathstrut +\mathstrut 27572162818669019136q^{48} \) \(\mathstrut +\mathstrut 330321418499425066617q^{49} \) \(\mathstrut +\mathstrut 744401203807562240000q^{50} \) \(\mathstrut -\mathstrut 260114896412659511928q^{51} \) \(\mathstrut +\mathstrut 1473777586764089655296q^{52} \) \(\mathstrut +\mathstrut 425070456489000653526q^{53} \) \(\mathstrut +\mathstrut 676061516644754227200q^{54} \) \(\mathstrut -\mathstrut 4946699438883545527800q^{55} \) \(\mathstrut +\mathstrut 2809433447516600270848q^{56} \) \(\mathstrut -\mathstrut 1371327646314598219440q^{57} \) \(\mathstrut -\mathstrut 8520623839647854223360q^{58} \) \(\mathstrut -\mathstrut 8338905271550069098980q^{59} \) \(\mathstrut +\mathstrut 560418071284128153600q^{60} \) \(\mathstrut +\mathstrut 24297948469063188136862q^{61} \) \(\mathstrut -\mathstrut 10909828592855757750272q^{62} \) \(\mathstrut +\mathstrut 34247108609328753153576q^{63} \) \(\mathstrut +\mathstrut 4722366482869645213696q^{64} \) \(\mathstrut +\mathstrut 29955270397463068982100q^{65} \) \(\mathstrut +\mathstrut 5820299342528476151808q^{66} \) \(\mathstrut -\mathstrut 124700073335102453318428q^{67} \) \(\mathstrut -\mathstrut 44550653374298805239808q^{68} \) \(\mathstrut +\mathstrut 8409794988603613209696q^{69} \) \(\mathstrut +\mathstrut 57103147272593075404800q^{70} \) \(\mathstrut -\mathstrut 93048982696907739984888q^{71} \) \(\mathstrut +\mathstrut 57565840534449948721152q^{72} \) \(\mathstrut +\mathstrut 40421798466182077528586q^{73} \) \(\mathstrut +\mathstrut 210450874287372026748928q^{74} \) \(\mathstrut -\mathstrut 17802383867229874702500q^{75} \) \(\mathstrut -\mathstrut 234871372136383869091840q^{76} \) \(\mathstrut +\mathstrut 593052629022873308571744q^{77} \) \(\mathstrut -\mathstrut 35245448556900187078656q^{78} \) \(\mathstrut -\mathstrut 805270113539155989394480q^{79} \) \(\mathstrut +\mathstrut 95984472949459098009600q^{80} \) \(\mathstrut +\mathstrut 693599895684166814041401q^{81} \) \(\mathstrut -\mathstrut 956660970636186955456512q^{82} \) \(\mathstrut +\mathstrut 8983348403447711887476q^{83} \) \(\mathstrut -\mathstrut 67187710640853539094528q^{84} \) \(\mathstrut -\mathstrut 905514427818742101513300q^{85} \) \(\mathstrut +\mathstrut 164387213700012497649664q^{86} \) \(\mathstrut +\mathstrut 203771051962047169995960q^{87} \) \(\mathstrut +\mathstrut 996860011170915824959488q^{88} \) \(\mathstrut +\mathstrut 3556004636909897335022490q^{89} \) \(\mathstrut +\mathstrut 1170054650276559103795200q^{90} \) \(\mathstrut -\mathstrut 3591293969199802514440208q^{91} \) \(\mathstrut +\mathstrut 1440370646407778565881856q^{92} \) \(\mathstrut +\mathstrut 260908976963324855025792q^{93} \) \(\mathstrut -\mathstrut 1146168914071583212634112q^{94} \) \(\mathstrut -\mathstrut 4773878721019496262909000q^{95} \) \(\mathstrut -\mathstrut 112935578905268302381056q^{96} \) \(\mathstrut -\mathstrut 8660489591283862898255518q^{97} \) \(\mathstrut -\mathstrut 1352996530173645072863232q^{98} \) \(\mathstrut +\mathstrut 12151764300038755701141756q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0
−4096.00 97956.0 1.67772e7 3.41005e8 −4.01228e8 −4.08826e10 −6.87195e10 −8.37693e11 −1.39676e12
\(n\): 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 \(T_{3} \) \(\mathstrut -\mathstrut 97956 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(2))\).