Properties

Label 312.2.a.f
Level 312
Weight 2
Character orbit 312.a
Self dual yes
Analytic conductor 2.491
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 312.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.49133254306\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 2q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} + 2q^{5} + q^{9} + q^{13} + 2q^{15} + 2q^{17} - 4q^{19} - q^{25} + q^{27} + 6q^{29} - 2q^{37} + q^{39} + 6q^{41} - 12q^{43} + 2q^{45} - 4q^{47} - 7q^{49} + 2q^{51} + 6q^{53} - 4q^{57} - 8q^{59} - 2q^{61} + 2q^{65} + 4q^{67} - 12q^{71} - 14q^{73} - q^{75} + q^{81} + 8q^{83} + 4q^{85} + 6q^{87} - 18q^{89} - 8q^{95} - 6q^{97} + O(q^{100}) \)

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 1.00000 0 2.00000 0 0 0 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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.a.f 1
3.b odd 2 1 936.2.a.b 1
4.b odd 2 1 624.2.a.d 1
5.b even 2 1 7800.2.a.d 1
8.b even 2 1 2496.2.a.c 1
8.d odd 2 1 2496.2.a.s 1
12.b even 2 1 1872.2.a.e 1
13.b even 2 1 4056.2.a.m 1
13.d odd 4 2 4056.2.c.h 2
24.f even 2 1 7488.2.a.br 1
24.h odd 2 1 7488.2.a.bs 1
52.b odd 2 1 8112.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.a.f 1 1.a even 1 1 trivial
624.2.a.d 1 4.b odd 2 1
936.2.a.b 1 3.b odd 2 1
1872.2.a.e 1 12.b even 2 1
2496.2.a.c 1 8.b even 2 1
2496.2.a.s 1 8.d odd 2 1
4056.2.a.m 1 13.b even 2 1
4056.2.c.h 2 13.d odd 4 2
7488.2.a.br 1 24.f even 2 1
7488.2.a.bs 1 24.h odd 2 1
7800.2.a.d 1 5.b even 2 1
8112.2.a.f 1 52.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(312))\):

\( T_{5} - 2 \)
\( T_{7} \)