Properties

Label 1176.2.a.e
Level $1176$
Weight $2$
Character orbit 1176.a
Self dual yes
Analytic conductor $9.390$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 2q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{5} + q^{9} - 6q^{11} + 3q^{13} - 2q^{15} - 4q^{17} + 5q^{19} - 4q^{23} - q^{25} + q^{27} - 4q^{29} - 7q^{31} - 6q^{33} - 9q^{37} + 3q^{39} + 2q^{41} - q^{43} - 2q^{45} - 2q^{47} - 4q^{51} + 8q^{53} + 12q^{55} + 5q^{57} - 10q^{61} - 6q^{65} - 15q^{67} - 4q^{69} - 6q^{71} + 11q^{73} - q^{75} + q^{79} + q^{81} - 6q^{83} + 8q^{85} - 4q^{87} + 8q^{89} - 7q^{93} - 10q^{95} + 14q^{97} - 6q^{99} + O(q^{100}) \)

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 1176.2.a.e 1
3.b odd 2 1 3528.2.a.y 1
4.b odd 2 1 2352.2.a.e 1
7.b odd 2 1 1176.2.a.d 1
7.c even 3 2 1176.2.q.e 2
7.d odd 6 2 168.2.q.b 2
8.b even 2 1 9408.2.a.bk 1
8.d odd 2 1 9408.2.a.cs 1
12.b even 2 1 7056.2.a.bn 1
21.c even 2 1 3528.2.a.f 1
21.g even 6 2 504.2.s.g 2
21.h odd 6 2 3528.2.s.d 2
28.d even 2 1 2352.2.a.x 1
28.f even 6 2 336.2.q.a 2
28.g odd 6 2 2352.2.q.v 2
56.e even 2 1 9408.2.a.f 1
56.h odd 2 1 9408.2.a.cd 1
56.j odd 6 2 1344.2.q.i 2
56.m even 6 2 1344.2.q.t 2
84.h odd 2 1 7056.2.a.i 1
84.j odd 6 2 1008.2.s.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.b 2 7.d odd 6 2
336.2.q.a 2 28.f even 6 2
504.2.s.g 2 21.g even 6 2
1008.2.s.m 2 84.j odd 6 2
1176.2.a.d 1 7.b odd 2 1
1176.2.a.e 1 1.a even 1 1 trivial
1176.2.q.e 2 7.c even 3 2
1344.2.q.i 2 56.j odd 6 2
1344.2.q.t 2 56.m even 6 2
2352.2.a.e 1 4.b odd 2 1
2352.2.a.x 1 28.d even 2 1
2352.2.q.v 2 28.g odd 6 2
3528.2.a.f 1 21.c even 2 1
3528.2.a.y 1 3.b odd 2 1
3528.2.s.d 2 21.h odd 6 2
7056.2.a.i 1 84.h odd 2 1
7056.2.a.bn 1 12.b even 2 1
9408.2.a.f 1 56.e even 2 1
9408.2.a.bk 1 8.b even 2 1
9408.2.a.cd 1 56.h odd 2 1
9408.2.a.cs 1 8.d odd 2 1

Hecke kernels

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

\( T_{5} + 2 \)
\( T_{11} + 6 \)
\( T_{13} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( 2 + T \)
$7$ \( T \)
$11$ \( 6 + T \)
$13$ \( -3 + T \)
$17$ \( 4 + T \)
$19$ \( -5 + T \)
$23$ \( 4 + T \)
$29$ \( 4 + T \)
$31$ \( 7 + T \)
$37$ \( 9 + T \)
$41$ \( -2 + T \)
$43$ \( 1 + T \)
$47$ \( 2 + T \)
$53$ \( -8 + T \)
$59$ \( T \)
$61$ \( 10 + T \)
$67$ \( 15 + T \)
$71$ \( 6 + T \)
$73$ \( -11 + T \)
$79$ \( -1 + T \)
$83$ \( 6 + T \)
$89$ \( -8 + T \)
$97$ \( -14 + T \)
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