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

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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

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

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

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