Properties

Label 9408.2.a.ck
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 1176)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + q^{9} + O(q^{10}) \) \( q + q^{3} + q^{9} - 4q^{13} + 4q^{17} - 4q^{19} - 4q^{23} - 5q^{25} + q^{27} - 2q^{29} - 8q^{31} + 6q^{37} - 4q^{39} + 12q^{41} + 4q^{43} + 8q^{47} + 4q^{51} - 6q^{53} - 4q^{57} + 12q^{59} - 4q^{61} - 4q^{67} - 4q^{69} + 12q^{71} + 8q^{73} - 5q^{75} + 16q^{79} + q^{81} - 4q^{83} - 2q^{87} + 4q^{89} - 8q^{93} + 16q^{97} + O(q^{100}) \)

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 0 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.ck 1
4.b odd 2 1 9408.2.a.u 1
7.b odd 2 1 9408.2.a.v 1
8.b even 2 1 2352.2.a.h 1
8.d odd 2 1 1176.2.a.h yes 1
24.f even 2 1 3528.2.a.n 1
24.h odd 2 1 7056.2.a.bc 1
28.d even 2 1 9408.2.a.cl 1
56.e even 2 1 1176.2.a.b 1
56.h odd 2 1 2352.2.a.r 1
56.j odd 6 2 2352.2.q.h 2
56.k odd 6 2 1176.2.q.c 2
56.m even 6 2 1176.2.q.h 2
56.p even 6 2 2352.2.q.t 2
168.e odd 2 1 3528.2.a.m 1
168.i even 2 1 7056.2.a.ba 1
168.v even 6 2 3528.2.s.n 2
168.be odd 6 2 3528.2.s.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.b 1 56.e even 2 1
1176.2.a.h yes 1 8.d odd 2 1
1176.2.q.c 2 56.k odd 6 2
1176.2.q.h 2 56.m even 6 2
2352.2.a.h 1 8.b even 2 1
2352.2.a.r 1 56.h odd 2 1
2352.2.q.h 2 56.j odd 6 2
2352.2.q.t 2 56.p even 6 2
3528.2.a.m 1 168.e odd 2 1
3528.2.a.n 1 24.f even 2 1
3528.2.s.m 2 168.be odd 6 2
3528.2.s.n 2 168.v even 6 2
7056.2.a.ba 1 168.i even 2 1
7056.2.a.bc 1 24.h odd 2 1
9408.2.a.u 1 4.b odd 2 1
9408.2.a.v 1 7.b odd 2 1
9408.2.a.ck 1 1.a even 1 1 trivial
9408.2.a.cl 1 28.d even 2 1

Hecke kernels

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

\( T_{5} \)
\( T_{11} \)
\( T_{13} + 4 \)
\( T_{17} - 4 \)
\( T_{19} + 4 \)
\( T_{31} + 8 \)