Properties

Label 4010.2.a.f
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

\(f(q)\) \(=\) \( q + q^{2} + 2q^{3} + q^{4} + q^{5} + 2q^{6} + 4q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + 2q^{3} + q^{4} + q^{5} + 2q^{6} + 4q^{7} + q^{8} + q^{9} + q^{10} - 4q^{11} + 2q^{12} + 4q^{13} + 4q^{14} + 2q^{15} + q^{16} - 4q^{17} + q^{18} + 4q^{19} + q^{20} + 8q^{21} - 4q^{22} + 6q^{23} + 2q^{24} + q^{25} + 4q^{26} - 4q^{27} + 4q^{28} - 6q^{29} + 2q^{30} + 8q^{31} + q^{32} - 8q^{33} - 4q^{34} + 4q^{35} + q^{36} - 4q^{37} + 4q^{38} + 8q^{39} + q^{40} - 10q^{41} + 8q^{42} + 4q^{43} - 4q^{44} + q^{45} + 6q^{46} + 8q^{47} + 2q^{48} + 9q^{49} + q^{50} - 8q^{51} + 4q^{52} - 12q^{53} - 4q^{54} - 4q^{55} + 4q^{56} + 8q^{57} - 6q^{58} + 12q^{59} + 2q^{60} - 10q^{61} + 8q^{62} + 4q^{63} + q^{64} + 4q^{65} - 8q^{66} - 2q^{67} - 4q^{68} + 12q^{69} + 4q^{70} + 4q^{71} + q^{72} + 10q^{73} - 4q^{74} + 2q^{75} + 4q^{76} - 16q^{77} + 8q^{78} - 8q^{79} + q^{80} - 11q^{81} - 10q^{82} - 4q^{83} + 8q^{84} - 4q^{85} + 4q^{86} - 12q^{87} - 4q^{88} - 6q^{89} + q^{90} + 16q^{91} + 6q^{92} + 16q^{93} + 8q^{94} + 4q^{95} + 2q^{96} + 8q^{97} + 9q^{98} - 4q^{99} + O(q^{100}) \)

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 4.00000 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3} - 2 \)
\( T_{7} - 4 \)
\( T_{11} + 4 \)