Properties

Label 1176.2.a.h
Level $1176$
Weight $2$
Character orbit 1176.a
Self dual yes
Analytic conductor $9.390$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.39040727770\)
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

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

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 1176.2.a.h yes 1
3.b odd 2 1 3528.2.a.n 1
4.b odd 2 1 2352.2.a.h 1
7.b odd 2 1 1176.2.a.b 1
7.c even 3 2 1176.2.q.c 2
7.d odd 6 2 1176.2.q.h 2
8.b even 2 1 9408.2.a.u 1
8.d odd 2 1 9408.2.a.ck 1
12.b even 2 1 7056.2.a.bc 1
21.c even 2 1 3528.2.a.m 1
21.g even 6 2 3528.2.s.m 2
21.h odd 6 2 3528.2.s.n 2
28.d even 2 1 2352.2.a.r 1
28.f even 6 2 2352.2.q.h 2
28.g odd 6 2 2352.2.q.t 2
56.e even 2 1 9408.2.a.v 1
56.h odd 2 1 9408.2.a.cl 1
84.h odd 2 1 7056.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.b 1 7.b odd 2 1
1176.2.a.h yes 1 1.a even 1 1 trivial
1176.2.q.c 2 7.c even 3 2
1176.2.q.h 2 7.d odd 6 2
2352.2.a.h 1 4.b odd 2 1
2352.2.a.r 1 28.d even 2 1
2352.2.q.h 2 28.f even 6 2
2352.2.q.t 2 28.g odd 6 2
3528.2.a.m 1 21.c even 2 1
3528.2.a.n 1 3.b odd 2 1
3528.2.s.m 2 21.g even 6 2
3528.2.s.n 2 21.h odd 6 2
7056.2.a.ba 1 84.h odd 2 1
7056.2.a.bc 1 12.b even 2 1
9408.2.a.u 1 8.b even 2 1
9408.2.a.v 1 56.e even 2 1
9408.2.a.ck 1 8.d odd 2 1
9408.2.a.cl 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\):

\( T_{5} \)
\( T_{11} \)
\( T_{13} - 4 \)