Properties

Label 4004.2.a.b
Level 4004
Weight 2
Character orbit 4004.a
Self dual Yes
Analytic conductor 31.972
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9721009693\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut +\mathstrut q^{13} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut +\mathstrut 5q^{23} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut +\mathstrut 5q^{27} \) \(\mathstrut +\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 3q^{31} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut +\mathstrut 3q^{35} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut q^{39} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 10q^{43} \) \(\mathstrut +\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut q^{49} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut -\mathstrut 10q^{53} \) \(\mathstrut +\mathstrut 3q^{55} \) \(\mathstrut -\mathstrut 5q^{59} \) \(\mathstrut +\mathstrut 8q^{61} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut 3q^{65} \) \(\mathstrut -\mathstrut 15q^{67} \) \(\mathstrut -\mathstrut 5q^{69} \) \(\mathstrut +\mathstrut q^{71} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 4q^{75} \) \(\mathstrut +\mathstrut q^{77} \) \(\mathstrut +\mathstrut q^{81} \) \(\mathstrut +\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 12q^{85} \) \(\mathstrut -\mathstrut 4q^{87} \) \(\mathstrut +\mathstrut 3q^{89} \) \(\mathstrut -\mathstrut q^{91} \) \(\mathstrut +\mathstrut 3q^{93} \) \(\mathstrut -\mathstrut 3q^{97} \) \(\mathstrut +\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0 −1.00000 0 −3.00000 0 −1.00000 0 −2.00000 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
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)
\(13\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).