Properties

Label 4013.2.a.a
Level 4013
Weight 2
Character orbit 4013.a
Self dual Yes
Analytic conductor 32.044
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4013 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4013.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0439663311\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} - 4q^{6} + q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} - 4q^{6} + q^{7} + q^{9} + 8q^{10} + 3q^{11} - 4q^{12} + 2q^{13} + 2q^{14} - 8q^{15} - 4q^{16} - 2q^{17} + 2q^{18} + q^{19} + 8q^{20} - 2q^{21} + 6q^{22} + 11q^{25} + 4q^{26} + 4q^{27} + 2q^{28} + 8q^{29} - 16q^{30} - 5q^{31} - 8q^{32} - 6q^{33} - 4q^{34} + 4q^{35} + 2q^{36} - 4q^{37} + 2q^{38} - 4q^{39} + 5q^{41} - 4q^{42} + 5q^{43} + 6q^{44} + 4q^{45} + 12q^{47} + 8q^{48} - 6q^{49} + 22q^{50} + 4q^{51} + 4q^{52} + q^{53} + 8q^{54} + 12q^{55} - 2q^{57} + 16q^{58} + 7q^{59} - 16q^{60} - 7q^{61} - 10q^{62} + q^{63} - 8q^{64} + 8q^{65} - 12q^{66} - 7q^{67} - 4q^{68} + 8q^{70} + 4q^{71} - 13q^{73} - 8q^{74} - 22q^{75} + 2q^{76} + 3q^{77} - 8q^{78} + 4q^{79} - 16q^{80} - 11q^{81} + 10q^{82} + 12q^{83} - 4q^{84} - 8q^{85} + 10q^{86} - 16q^{87} - 10q^{89} + 8q^{90} + 2q^{91} + 10q^{93} + 24q^{94} + 4q^{95} + 16q^{96} - 17q^{97} - 12q^{98} + 3q^{99} + O(q^{100}) \)

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
0
2.00000 −2.00000 2.00000 4.00000 −4.00000 1.00000 0 1.00000 8.00000
\(n\): 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
\(4013\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4013))\).