Properties

Label 315.2.a.b
Level 315
Weight 2
Character orbit 315.a
Self dual Yes
Analytic conductor 2.515
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.51528766367\)
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

\(f(q)\) \(=\) \( q - 2q^{4} + q^{5} + q^{7} + O(q^{10}) \) \( q - 2q^{4} + q^{5} + q^{7} + 3q^{11} + 5q^{13} + 4q^{16} - 3q^{17} + 2q^{19} - 2q^{20} + 6q^{23} + q^{25} - 2q^{28} - 3q^{29} - 4q^{31} + q^{35} + 2q^{37} + 12q^{41} - 10q^{43} - 6q^{44} - 9q^{47} + q^{49} - 10q^{52} - 12q^{53} + 3q^{55} + 8q^{61} - 8q^{64} + 5q^{65} - 4q^{67} + 6q^{68} + 2q^{73} - 4q^{76} + 3q^{77} - q^{79} + 4q^{80} - 12q^{83} - 3q^{85} + 12q^{89} + 5q^{91} - 12q^{92} + 2q^{95} - q^{97} + O(q^{100}) \)

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
0 0 −2.00000 1.00000 0 1.00000 0 0 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
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\).