Properties

Label 3038.2.a.j
Level 3038
Weight 2
Character orbit 3038.a
Self dual Yes
Analytic conductor 24.259
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3038 = 2 \cdot 7^{2} \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3038.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(24.2585521341\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 2q^{5} + q^{8} - 3q^{9} + O(q^{10}) \) \( q + q^{2} + q^{4} + 2q^{5} + q^{8} - 3q^{9} + 2q^{10} - 2q^{13} + q^{16} + 6q^{17} - 3q^{18} - 4q^{19} + 2q^{20} + 8q^{23} - q^{25} - 2q^{26} + 2q^{29} + q^{31} + q^{32} + 6q^{34} - 3q^{36} + 10q^{37} - 4q^{38} + 2q^{40} + 6q^{41} + 8q^{43} - 6q^{45} + 8q^{46} + 8q^{47} - q^{50} - 2q^{52} - 6q^{53} + 2q^{58} + 12q^{59} + 6q^{61} + q^{62} + q^{64} - 4q^{65} - 12q^{67} + 6q^{68} + 8q^{71} - 3q^{72} - 10q^{73} + 10q^{74} - 4q^{76} - 8q^{79} + 2q^{80} + 9q^{81} + 6q^{82} - 8q^{83} + 12q^{85} + 8q^{86} + 6q^{89} - 6q^{90} + 8q^{92} + 8q^{94} - 8q^{95} - 2q^{97} + O(q^{100}) \)

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 2.00000 0 0 1.00000 −3.00000 2.00000
\(n\): 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
\(2\) \(-1\)
\(7\) \(-1\)
\(31\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3038))\):

\( T_{3} \)
\( T_{5} - 2 \)
\( T_{11} \)
\( T_{13} + 2 \)