Properties

Label 4002.2.a.b
Level 4002
Weight 2
Character orbit 4002.a
Self dual Yes
Analytic conductor 31.956
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4002.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9561308889\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut -\mathstrut q^{12} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut q^{16} \) \(\mathstrut -\mathstrut q^{18} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut q^{24} \) \(\mathstrut -\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 6q^{26} \) \(\mathstrut -\mathstrut q^{27} \) \(\mathstrut -\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut q^{29} \) \(\mathstrut -\mathstrut 8q^{31} \) \(\mathstrut -\mathstrut q^{32} \) \(\mathstrut +\mathstrut q^{36} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 4q^{42} \) \(\mathstrut +\mathstrut q^{46} \) \(\mathstrut -\mathstrut q^{48} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut +\mathstrut 5q^{50} \) \(\mathstrut +\mathstrut 6q^{52} \) \(\mathstrut +\mathstrut q^{54} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut +\mathstrut q^{58} \) \(\mathstrut +\mathstrut 6q^{61} \) \(\mathstrut +\mathstrut 8q^{62} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut q^{64} \) \(\mathstrut -\mathstrut 10q^{67} \) \(\mathstrut +\mathstrut q^{69} \) \(\mathstrut +\mathstrut 16q^{71} \) \(\mathstrut -\mathstrut q^{72} \) \(\mathstrut +\mathstrut 6q^{73} \) \(\mathstrut -\mathstrut 2q^{74} \) \(\mathstrut +\mathstrut 5q^{75} \) \(\mathstrut +\mathstrut 6q^{78} \) \(\mathstrut +\mathstrut 10q^{79} \) \(\mathstrut +\mathstrut q^{81} \) \(\mathstrut -\mathstrut 10q^{82} \) \(\mathstrut +\mathstrut 6q^{83} \) \(\mathstrut +\mathstrut 4q^{84} \) \(\mathstrut +\mathstrut q^{87} \) \(\mathstrut -\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut 24q^{91} \) \(\mathstrut -\mathstrut q^{92} \) \(\mathstrut +\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut q^{96} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 9q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −1.00000 1.00000 0 1.00000 −4.00000 −1.00000 1.00000 0
\(n\): 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
\(2\) \(1\)
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5} \)
\(T_{7} \) \(\mathstrut +\mathstrut 4 \)
\(T_{11} \)