Properties

Label 9408.2.a.bh
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4704)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 2q^{5} + q^{9} + O(q^{10}) \) \( q - q^{3} + 2q^{5} + q^{9} + 2q^{11} - 2q^{15} - 2q^{17} + 2q^{23} - q^{25} - q^{27} - 6q^{29} + 4q^{31} - 2q^{33} - 6q^{37} + 2q^{41} + 2q^{45} + 2q^{51} + 6q^{53} + 4q^{55} + 12q^{59} + 12q^{61} + 12q^{67} - 2q^{69} - 10q^{71} - 12q^{73} + q^{75} + 12q^{79} + q^{81} + 12q^{83} - 4q^{85} + 6q^{87} - 14q^{89} - 4q^{93} + 12q^{97} + 2q^{99} + O(q^{100}) \)

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:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)

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 9408.2.a.bh 1
4.b odd 2 1 9408.2.a.cw 1
7.b odd 2 1 9408.2.a.by 1
8.b even 2 1 4704.2.a.u yes 1
8.d odd 2 1 4704.2.a.d 1
28.d even 2 1 9408.2.a.j 1
56.e even 2 1 4704.2.a.bd yes 1
56.h odd 2 1 4704.2.a.n yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.d 1 8.d odd 2 1
4704.2.a.n yes 1 56.h odd 2 1
4704.2.a.u yes 1 8.b even 2 1
4704.2.a.bd yes 1 56.e even 2 1
9408.2.a.j 1 28.d even 2 1
9408.2.a.bh 1 1.a even 1 1 trivial
9408.2.a.by 1 7.b odd 2 1
9408.2.a.cw 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5} - 2 \)
\( T_{11} - 2 \)
\( T_{13} \)
\( T_{17} + 2 \)
\( T_{19} \)
\( T_{31} - 4 \)