Properties

Label 1176.2.a.i
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: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 2q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} + 2q^{5} + q^{9} + 4q^{11} + 2q^{13} + 2q^{15} - 2q^{17} + 4q^{19} - 8q^{23} - q^{25} + q^{27} + 6q^{29} - 8q^{31} + 4q^{33} + 6q^{37} + 2q^{39} + 6q^{41} + 4q^{43} + 2q^{45} - 2q^{51} - 2q^{53} + 8q^{55} + 4q^{57} - 4q^{59} + 2q^{61} + 4q^{65} - 4q^{67} - 8q^{69} + 8q^{71} - 10q^{73} - q^{75} - 8q^{79} + q^{81} + 4q^{83} - 4q^{85} + 6q^{87} + 6q^{89} - 8q^{93} + 8q^{95} - 2q^{97} + 4q^{99} + O(q^{100}) \)

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.i 1
3.b odd 2 1 3528.2.a.d 1
4.b odd 2 1 2352.2.a.i 1
7.b odd 2 1 24.2.a.a 1
7.c even 3 2 1176.2.q.a 2
7.d odd 6 2 1176.2.q.i 2
8.b even 2 1 9408.2.a.h 1
8.d odd 2 1 9408.2.a.cc 1
12.b even 2 1 7056.2.a.q 1
21.c even 2 1 72.2.a.a 1
21.g even 6 2 3528.2.s.j 2
21.h odd 6 2 3528.2.s.y 2
28.d even 2 1 48.2.a.a 1
28.f even 6 2 2352.2.q.l 2
28.g odd 6 2 2352.2.q.r 2
35.c odd 2 1 600.2.a.h 1
35.f even 4 2 600.2.f.e 2
56.e even 2 1 192.2.a.b 1
56.h odd 2 1 192.2.a.d 1
63.l odd 6 2 648.2.i.g 2
63.o even 6 2 648.2.i.b 2
77.b even 2 1 2904.2.a.c 1
84.h odd 2 1 144.2.a.b 1
91.b odd 2 1 4056.2.a.i 1
91.i even 4 2 4056.2.c.e 2
105.g even 2 1 1800.2.a.m 1
105.k odd 4 2 1800.2.f.c 2
112.j even 4 2 768.2.d.d 2
112.l odd 4 2 768.2.d.e 2
119.d odd 2 1 6936.2.a.p 1
133.c even 2 1 8664.2.a.j 1
140.c even 2 1 1200.2.a.d 1
140.j odd 4 2 1200.2.f.b 2
168.e odd 2 1 576.2.a.b 1
168.i even 2 1 576.2.a.d 1
231.h odd 2 1 8712.2.a.u 1
252.s odd 6 2 1296.2.i.e 2
252.bi even 6 2 1296.2.i.m 2
280.c odd 2 1 4800.2.a.q 1
280.n even 2 1 4800.2.a.cc 1
280.s even 4 2 4800.2.f.d 2
280.y odd 4 2 4800.2.f.bg 2
308.g odd 2 1 5808.2.a.s 1
336.v odd 4 2 2304.2.d.k 2
336.y even 4 2 2304.2.d.i 2
364.h even 2 1 8112.2.a.be 1
420.o odd 2 1 3600.2.a.v 1
420.w even 4 2 3600.2.f.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 7.b odd 2 1
48.2.a.a 1 28.d even 2 1
72.2.a.a 1 21.c even 2 1
144.2.a.b 1 84.h odd 2 1
192.2.a.b 1 56.e even 2 1
192.2.a.d 1 56.h odd 2 1
576.2.a.b 1 168.e odd 2 1
576.2.a.d 1 168.i even 2 1
600.2.a.h 1 35.c odd 2 1
600.2.f.e 2 35.f even 4 2
648.2.i.b 2 63.o even 6 2
648.2.i.g 2 63.l odd 6 2
768.2.d.d 2 112.j even 4 2
768.2.d.e 2 112.l odd 4 2
1176.2.a.i 1 1.a even 1 1 trivial
1176.2.q.a 2 7.c even 3 2
1176.2.q.i 2 7.d odd 6 2
1200.2.a.d 1 140.c even 2 1
1200.2.f.b 2 140.j odd 4 2
1296.2.i.e 2 252.s odd 6 2
1296.2.i.m 2 252.bi even 6 2
1800.2.a.m 1 105.g even 2 1
1800.2.f.c 2 105.k odd 4 2
2304.2.d.i 2 336.y even 4 2
2304.2.d.k 2 336.v odd 4 2
2352.2.a.i 1 4.b odd 2 1
2352.2.q.l 2 28.f even 6 2
2352.2.q.r 2 28.g odd 6 2
2904.2.a.c 1 77.b even 2 1
3528.2.a.d 1 3.b odd 2 1
3528.2.s.j 2 21.g even 6 2
3528.2.s.y 2 21.h odd 6 2
3600.2.a.v 1 420.o odd 2 1
3600.2.f.r 2 420.w even 4 2
4056.2.a.i 1 91.b odd 2 1
4056.2.c.e 2 91.i even 4 2
4800.2.a.q 1 280.c odd 2 1
4800.2.a.cc 1 280.n even 2 1
4800.2.f.d 2 280.s even 4 2
4800.2.f.bg 2 280.y odd 4 2
5808.2.a.s 1 308.g odd 2 1
6936.2.a.p 1 119.d odd 2 1
7056.2.a.q 1 12.b even 2 1
8112.2.a.be 1 364.h even 2 1
8664.2.a.j 1 133.c even 2 1
8712.2.a.u 1 231.h odd 2 1
9408.2.a.h 1 8.b even 2 1
9408.2.a.cc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\):

\( T_{5} - 2 \)
\( T_{11} - 4 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

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