Newform orbit 1176.1.s.c

• Label: 1176.1.s.c
• Level: $1176$
• Weight: $1$
• Character orbit: 1176.s
• Analytic conductor: $0.587$
• Analytic rank: $0$
• Dimension: $8$
• Projective image: $D_{4}$
• CM discriminant: -56
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1176.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.586900454856\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.2.4032.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q - \zeta_{24}^{10} q^{2} - \zeta_{24}^{5} q^{3} - \zeta_{24}^{8} q^{4} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{5} - \zeta_{24}^{3} q^{6} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{10} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4}+O(q^{10}) \)

Character values

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

\(n\)	\(295\)	\(589\)	\(785\)	\(1081\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-\zeta_{24}^{4}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

557.1

0.258819	−	0.965926i
−0.258819	+	0.965926i
0.965926	+	0.258819i
−0.965926	−	0.258819i
0.258819	+	0.965926i
−0.258819	−	0.965926i
0.965926	−	0.258819i
−0.965926	+	0.258819i

−0.866025

+

0.500000i

−0.965926

+

0.258819i

0.500000

−

0.866025i

0.707107

+

1.22474i

0.707107

−

0.707107i

0

1.00000i

0.866025

−

0.500000i

−1.22474

−

0.707107i

557.2

−0.866025

+

0.500000i

0.965926

−

0.258819i

0.500000

−

0.866025i

−0.707107

−

1.22474i

−0.707107

+

0.707107i

0

1.00000i

0.866025

−

0.500000i

1.22474

+

0.707107i

557.3

0.866025

−

0.500000i

−0.258819

−

0.965926i

0.500000

−

0.866025i

−0.707107

−

1.22474i

−0.707107

−

0.707107i

0

−

1.00000i

−0.866025

+

0.500000i

−1.22474

−

0.707107i

557.4

0.866025

−

0.500000i

0.258819

+

0.965926i

0.500000

−

0.866025i

0.707107

+

1.22474i

0.707107

+

0.707107i

0

−

1.00000i

−0.866025

+

0.500000i

1.22474

+

0.707107i

1157.1

−0.866025

−

0.500000i

−0.965926

−

0.258819i

0.500000

+

0.866025i

0.707107

−

1.22474i

0.707107

+

0.707107i

0

−

1.00000i

0.866025

+

0.500000i

−1.22474

+

0.707107i

1157.2

−0.866025

−

0.500000i

0.965926

+

0.258819i

0.500000

+

0.866025i

−0.707107

+

1.22474i

−0.707107

−

0.707107i

0

−

1.00000i

0.866025

+

0.500000i

1.22474

−

0.707107i

1157.3

0.866025

+

0.500000i

−0.258819

+

0.965926i

0.500000

+

0.866025i

−0.707107

+

1.22474i

−0.707107

+

0.707107i

0

1.00000i

−0.866025

−

0.500000i

−1.22474

+

0.707107i

1157.4

0.866025

+

0.500000i

0.258819

−

0.965926i

0.500000

+

0.866025i

0.707107

−

1.22474i

0.707107

−

0.707107i

0

1.00000i

−0.866025

−

0.500000i

1.22474

−

0.707107i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
56.h	odd	2	1	CM by \(\Q(\sqrt{-14}) \)
3.b	odd	2	1	inner
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
8.b	even	2	1	inner
21.c	even	2	1	inner
21.g	even	6	1	inner
21.h	odd	6	1	inner
24.h	odd	2	1	inner
56.j	odd	6	1	inner
56.p	even	6	1	inner
168.i	even	2	1	inner
168.s	odd	6	1	inner
168.ba	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1176.1.s.c		8
3.b	odd	2	1	inner	1176.1.s.c		8
7.b	odd	2	1	inner	1176.1.s.c		8
7.c	even	3	1		1176.1.n.e	✓	4
7.c	even	3	1	inner	1176.1.s.c		8
7.d	odd	6	1		1176.1.n.e	✓	4
7.d	odd	6	1	inner	1176.1.s.c		8
8.b	even	2	1	inner	1176.1.s.c		8
21.c	even	2	1	inner	1176.1.s.c		8
21.g	even	6	1		1176.1.n.e	✓	4
21.g	even	6	1	inner	1176.1.s.c		8
21.h	odd	6	1		1176.1.n.e	✓	4
21.h	odd	6	1	inner	1176.1.s.c		8
24.h	odd	2	1	inner	1176.1.s.c		8
56.h	odd	2	1	CM	1176.1.s.c		8
56.j	odd	6	1		1176.1.n.e	✓	4
56.j	odd	6	1	inner	1176.1.s.c		8
56.p	even	6	1		1176.1.n.e	✓	4
56.p	even	6	1	inner	1176.1.s.c		8
168.i	even	2	1	inner	1176.1.s.c		8
168.s	odd	6	1		1176.1.n.e	✓	4
168.s	odd	6	1	inner	1176.1.s.c		8
168.ba	even	6	1		1176.1.n.e	✓	4
168.ba	even	6	1	inner	1176.1.s.c		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1176.1.n.e	✓	4	7.c	even	3	1
1176.1.n.e	✓	4	7.d	odd	6	1
1176.1.n.e	✓	4	21.g	even	6	1
1176.1.n.e	✓	4	21.h	odd	6	1
1176.1.n.e	✓	4	56.j	odd	6	1
1176.1.n.e	✓	4	56.p	even	6	1
1176.1.n.e	✓	4	168.s	odd	6	1
1176.1.n.e	✓	4	168.ba	even	6	1
1176.1.s.c		8	1.a	even	1	1	trivial
1176.1.s.c		8	3.b	odd	2	1	inner
1176.1.s.c		8	7.b	odd	2	1	inner
1176.1.s.c		8	7.c	even	3	1	inner
1176.1.s.c		8	7.d	odd	6	1	inner
1176.1.s.c		8	8.b	even	2	1	inner
1176.1.s.c		8	21.c	even	2	1	inner
1176.1.s.c		8	21.g	even	6	1	inner
1176.1.s.c		8	21.h	odd	6	1	inner
1176.1.s.c		8	24.h	odd	2	1	inner
1176.1.s.c		8	56.h	odd	2	1	CM
1176.1.s.c		8	56.j	odd	6	1	inner
1176.1.s.c		8	56.p	even	6	1	inner
1176.1.s.c		8	168.i	even	2	1	inner
1176.1.s.c		8	168.s	odd	6	1	inner
1176.1.s.c		8	168.ba	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2T_{5}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1176, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{2} + 1)^{2} \)
$3$	\( T^{8} - T^{4} + 1 \)
$5$	\( (T^{4} + 2 T^{2} + 4)^{2} \)
$7$	\( T^{8} \)
$11$	\( T^{8} \)