Properties

Label 4010.2.a.b
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 1
Dimension 1
CM No
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0200112105\)
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

\(f(q)\) \(=\) \( q - q^{2} + 2q^{3} + q^{4} - q^{5} - 2q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + 2q^{3} + q^{4} - q^{5} - 2q^{6} - q^{8} + q^{9} + q^{10} + 3q^{11} + 2q^{12} + q^{13} - 2q^{15} + q^{16} - 8q^{17} - q^{18} - q^{20} - 3q^{22} - 2q^{24} + q^{25} - q^{26} - 4q^{27} - 4q^{29} + 2q^{30} + q^{31} - q^{32} + 6q^{33} + 8q^{34} + q^{36} - 10q^{37} + 2q^{39} + q^{40} + 7q^{41} - 11q^{43} + 3q^{44} - q^{45} + 4q^{47} + 2q^{48} - 7q^{49} - q^{50} - 16q^{51} + q^{52} - 5q^{53} + 4q^{54} - 3q^{55} + 4q^{58} + 4q^{59} - 2q^{60} - 13q^{61} - q^{62} + q^{64} - q^{65} - 6q^{66} - 8q^{68} + 9q^{71} - q^{72} + 9q^{73} + 10q^{74} + 2q^{75} - 2q^{78} + 3q^{79} - q^{80} - 11q^{81} - 7q^{82} - 5q^{83} + 8q^{85} + 11q^{86} - 8q^{87} - 3q^{88} + 7q^{89} + q^{90} + 2q^{93} - 4q^{94} - 2q^{96} - 18q^{97} + 7q^{98} + 3q^{99} + O(q^{100}) \)

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
−1.00000 2.00000 1.00000 −1.00000 −2.00000 0 −1.00000 1.00000 1.00000
\(n\): 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
\(2\) \(1\)
\(5\) \(1\)
\(401\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3} - 2 \)
\( T_{7} \)
\( T_{11} - 3 \)