Properties

Label 1176.1.n.c
Level $1176$
Weight $1$
Character orbit 1176.n
Self dual yes
Analytic conductor $0.587$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -24
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1176.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.586900454856\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1176.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.9680832.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} - q^{11} - q^{12} - q^{15} + q^{16} + q^{18} + q^{20} - q^{22} - q^{24} - q^{27} - q^{29} - q^{30} + q^{31} + q^{32} + q^{33} + q^{36} + q^{40} - q^{44} + q^{45} - q^{48} - q^{53} - q^{54} - q^{55} - q^{58} + q^{59} - q^{60} + q^{62} + q^{64} + q^{66} + q^{72} - 2 q^{73} - q^{79} + q^{80} + q^{81} + q^{83} + q^{87} - q^{88} + q^{90} - q^{93} - q^{96} + q^{97} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
0
1.00000 −1.00000 1.00000 1.00000 −1.00000 0 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.1.n.c 1
3.b odd 2 1 1176.1.n.b 1
7.b odd 2 1 1176.1.n.d 1
7.c even 3 2 1176.1.s.a 2
7.d odd 6 2 168.1.s.a 2
8.b even 2 1 1176.1.n.b 1
21.c even 2 1 1176.1.n.a 1
21.g even 6 2 168.1.s.b yes 2
21.h odd 6 2 1176.1.s.b 2
24.h odd 2 1 CM 1176.1.n.c 1
28.f even 6 2 672.1.ba.b 2
56.h odd 2 1 1176.1.n.a 1
56.j odd 6 2 168.1.s.b yes 2
56.m even 6 2 672.1.ba.a 2
56.p even 6 2 1176.1.s.b 2
84.j odd 6 2 672.1.ba.a 2
168.i even 2 1 1176.1.n.d 1
168.s odd 6 2 1176.1.s.a 2
168.ba even 6 2 168.1.s.a 2
168.be odd 6 2 672.1.ba.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.1.s.a 2 7.d odd 6 2
168.1.s.a 2 168.ba even 6 2
168.1.s.b yes 2 21.g even 6 2
168.1.s.b yes 2 56.j odd 6 2
672.1.ba.a 2 56.m even 6 2
672.1.ba.a 2 84.j odd 6 2
672.1.ba.b 2 28.f even 6 2
672.1.ba.b 2 168.be odd 6 2
1176.1.n.a 1 21.c even 2 1
1176.1.n.a 1 56.h odd 2 1
1176.1.n.b 1 3.b odd 2 1
1176.1.n.b 1 8.b even 2 1
1176.1.n.c 1 1.a even 1 1 trivial
1176.1.n.c 1 24.h odd 2 1 CM
1176.1.n.d 1 7.b odd 2 1
1176.1.n.d 1 168.i even 2 1
1176.1.s.a 2 7.c even 3 2
1176.1.s.a 2 168.s odd 6 2
1176.1.s.b 2 21.h odd 6 2
1176.1.s.b 2 56.p even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1176, [\chi])\):

\( T_{5} - 1 \)
\( T_{11} + 1 \)

Hecke characteristic polynomials

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