Properties

Label 4010.2.a.d
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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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: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} + 3q^{3} + q^{4} - q^{5} - 3q^{6} + q^{7} - q^{8} + 6q^{9} + O(q^{10}) \) \( q - q^{2} + 3q^{3} + q^{4} - q^{5} - 3q^{6} + q^{7} - q^{8} + 6q^{9} + q^{10} + 5q^{11} + 3q^{12} + 4q^{13} - q^{14} - 3q^{15} + q^{16} + 5q^{17} - 6q^{18} + 4q^{19} - q^{20} + 3q^{21} - 5q^{22} - 2q^{23} - 3q^{24} + q^{25} - 4q^{26} + 9q^{27} + q^{28} + 2q^{29} + 3q^{30} - 7q^{31} - q^{32} + 15q^{33} - 5q^{34} - q^{35} + 6q^{36} - 8q^{37} - 4q^{38} + 12q^{39} + q^{40} + 2q^{41} - 3q^{42} + 5q^{44} - 6q^{45} + 2q^{46} + 3q^{48} - 6q^{49} - q^{50} + 15q^{51} + 4q^{52} - 2q^{53} - 9q^{54} - 5q^{55} - q^{56} + 12q^{57} - 2q^{58} - 6q^{59} - 3q^{60} - 2q^{61} + 7q^{62} + 6q^{63} + q^{64} - 4q^{65} - 15q^{66} + 7q^{67} + 5q^{68} - 6q^{69} + q^{70} - 13q^{71} - 6q^{72} - 6q^{73} + 8q^{74} + 3q^{75} + 4q^{76} + 5q^{77} - 12q^{78} - q^{80} + 9q^{81} - 2q^{82} + 6q^{83} + 3q^{84} - 5q^{85} + 6q^{87} - 5q^{88} + 6q^{89} + 6q^{90} + 4q^{91} - 2q^{92} - 21q^{93} - 4q^{95} - 3q^{96} - q^{97} + 6q^{98} + 30q^{99} + O(q^{100}) \)

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 3.00000 1.00000 −1.00000 −3.00000 1.00000 −1.00000 6.00000 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3} - 3 \)
\( T_{7} - 1 \)
\( T_{11} - 5 \)