Properties

Label 8002.2.a.b
Level 8002
Weight 2
Character orbit 8002.a
Self dual Yes
Analytic conductor 63.896
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8002.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.8962916974\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut -\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut -\mathstrut 3q^{9} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut q^{16} \) \(\mathstrut -\mathstrut 3q^{18} \) \(\mathstrut +\mathstrut q^{19} \) \(\mathstrut -\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut 3q^{23} \) \(\mathstrut -\mathstrut 5q^{25} \) \(\mathstrut +\mathstrut 4q^{26} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut q^{32} \) \(\mathstrut -\mathstrut 3q^{36} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut q^{38} \) \(\mathstrut -\mathstrut 8q^{41} \) \(\mathstrut -\mathstrut 12q^{43} \) \(\mathstrut -\mathstrut 2q^{44} \) \(\mathstrut +\mathstrut 3q^{46} \) \(\mathstrut -\mathstrut 10q^{47} \) \(\mathstrut -\mathstrut 7q^{49} \) \(\mathstrut -\mathstrut 5q^{50} \) \(\mathstrut +\mathstrut 4q^{52} \) \(\mathstrut -\mathstrut 13q^{53} \) \(\mathstrut -\mathstrut 4q^{58} \) \(\mathstrut +\mathstrut 7q^{59} \) \(\mathstrut +\mathstrut 12q^{61} \) \(\mathstrut +\mathstrut q^{64} \) \(\mathstrut -\mathstrut 16q^{67} \) \(\mathstrut +\mathstrut 8q^{71} \) \(\mathstrut -\mathstrut 3q^{72} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut +\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut q^{76} \) \(\mathstrut +\mathstrut 10q^{79} \) \(\mathstrut +\mathstrut 9q^{81} \) \(\mathstrut -\mathstrut 8q^{82} \) \(\mathstrut -\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 12q^{86} \) \(\mathstrut -\mathstrut 2q^{88} \) \(\mathstrut +\mathstrut 15q^{89} \) \(\mathstrut +\mathstrut 3q^{92} \) \(\mathstrut -\mathstrut 10q^{94} \) \(\mathstrut +\mathstrut 10q^{97} \) \(\mathstrut -\mathstrut 7q^{98} \) \(\mathstrut +\mathstrut 6q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 0 1.00000 0 0 0 1.00000 −3.00000 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
\(2\) \(-1\)
\(4001\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\).