Properties

Label 13.4.a.a
Level 13
Weight 4
Character orbit 13.a
Self dual Yes
Analytic conductor 0.767
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 13.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.767024830075\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 35q^{6} \) \(\mathstrut -\mathstrut 13q^{7} \) \(\mathstrut -\mathstrut 45q^{8} \) \(\mathstrut +\mathstrut 22q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 35q^{6} \) \(\mathstrut -\mathstrut 13q^{7} \) \(\mathstrut -\mathstrut 45q^{8} \) \(\mathstrut +\mathstrut 22q^{9} \) \(\mathstrut +\mathstrut 35q^{10} \) \(\mathstrut -\mathstrut 26q^{11} \) \(\mathstrut -\mathstrut 119q^{12} \) \(\mathstrut +\mathstrut 13q^{13} \) \(\mathstrut +\mathstrut 65q^{14} \) \(\mathstrut +\mathstrut 49q^{15} \) \(\mathstrut +\mathstrut 89q^{16} \) \(\mathstrut +\mathstrut 77q^{17} \) \(\mathstrut -\mathstrut 110q^{18} \) \(\mathstrut -\mathstrut 126q^{19} \) \(\mathstrut -\mathstrut 119q^{20} \) \(\mathstrut +\mathstrut 91q^{21} \) \(\mathstrut +\mathstrut 130q^{22} \) \(\mathstrut -\mathstrut 96q^{23} \) \(\mathstrut +\mathstrut 315q^{24} \) \(\mathstrut -\mathstrut 76q^{25} \) \(\mathstrut -\mathstrut 65q^{26} \) \(\mathstrut +\mathstrut 35q^{27} \) \(\mathstrut -\mathstrut 221q^{28} \) \(\mathstrut -\mathstrut 82q^{29} \) \(\mathstrut -\mathstrut 245q^{30} \) \(\mathstrut +\mathstrut 196q^{31} \) \(\mathstrut -\mathstrut 85q^{32} \) \(\mathstrut +\mathstrut 182q^{33} \) \(\mathstrut -\mathstrut 385q^{34} \) \(\mathstrut +\mathstrut 91q^{35} \) \(\mathstrut +\mathstrut 374q^{36} \) \(\mathstrut -\mathstrut 131q^{37} \) \(\mathstrut +\mathstrut 630q^{38} \) \(\mathstrut -\mathstrut 91q^{39} \) \(\mathstrut +\mathstrut 315q^{40} \) \(\mathstrut +\mathstrut 336q^{41} \) \(\mathstrut -\mathstrut 455q^{42} \) \(\mathstrut -\mathstrut 201q^{43} \) \(\mathstrut -\mathstrut 442q^{44} \) \(\mathstrut -\mathstrut 154q^{45} \) \(\mathstrut +\mathstrut 480q^{46} \) \(\mathstrut -\mathstrut 105q^{47} \) \(\mathstrut -\mathstrut 623q^{48} \) \(\mathstrut -\mathstrut 174q^{49} \) \(\mathstrut +\mathstrut 380q^{50} \) \(\mathstrut -\mathstrut 539q^{51} \) \(\mathstrut +\mathstrut 221q^{52} \) \(\mathstrut -\mathstrut 432q^{53} \) \(\mathstrut -\mathstrut 175q^{54} \) \(\mathstrut +\mathstrut 182q^{55} \) \(\mathstrut +\mathstrut 585q^{56} \) \(\mathstrut +\mathstrut 882q^{57} \) \(\mathstrut +\mathstrut 410q^{58} \) \(\mathstrut -\mathstrut 294q^{59} \) \(\mathstrut +\mathstrut 833q^{60} \) \(\mathstrut -\mathstrut 56q^{61} \) \(\mathstrut -\mathstrut 980q^{62} \) \(\mathstrut -\mathstrut 286q^{63} \) \(\mathstrut -\mathstrut 287q^{64} \) \(\mathstrut -\mathstrut 91q^{65} \) \(\mathstrut -\mathstrut 910q^{66} \) \(\mathstrut +\mathstrut 478q^{67} \) \(\mathstrut +\mathstrut 1309q^{68} \) \(\mathstrut +\mathstrut 672q^{69} \) \(\mathstrut -\mathstrut 455q^{70} \) \(\mathstrut +\mathstrut 9q^{71} \) \(\mathstrut -\mathstrut 990q^{72} \) \(\mathstrut +\mathstrut 98q^{73} \) \(\mathstrut +\mathstrut 655q^{74} \) \(\mathstrut +\mathstrut 532q^{75} \) \(\mathstrut -\mathstrut 2142q^{76} \) \(\mathstrut +\mathstrut 338q^{77} \) \(\mathstrut +\mathstrut 455q^{78} \) \(\mathstrut +\mathstrut 1304q^{79} \) \(\mathstrut -\mathstrut 623q^{80} \) \(\mathstrut -\mathstrut 839q^{81} \) \(\mathstrut -\mathstrut 1680q^{82} \) \(\mathstrut -\mathstrut 308q^{83} \) \(\mathstrut +\mathstrut 1547q^{84} \) \(\mathstrut -\mathstrut 539q^{85} \) \(\mathstrut +\mathstrut 1005q^{86} \) \(\mathstrut +\mathstrut 574q^{87} \) \(\mathstrut +\mathstrut 1170q^{88} \) \(\mathstrut -\mathstrut 1190q^{89} \) \(\mathstrut +\mathstrut 770q^{90} \) \(\mathstrut -\mathstrut 169q^{91} \) \(\mathstrut -\mathstrut 1632q^{92} \) \(\mathstrut -\mathstrut 1372q^{93} \) \(\mathstrut +\mathstrut 525q^{94} \) \(\mathstrut +\mathstrut 882q^{95} \) \(\mathstrut +\mathstrut 595q^{96} \) \(\mathstrut +\mathstrut 70q^{97} \) \(\mathstrut +\mathstrut 870q^{98} \) \(\mathstrut -\mathstrut 572q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−5.00000 −7.00000 17.0000 −7.00000 35.0000 −13.0000 −45.0000 22.0000 35.0000
\(n\): 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
\(13\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut +\mathstrut 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\).