Newform orbit 1156.1.g.a

• Label: 1156.1.g.a
• Level: $1156$
• Weight: $1$
• Character orbit: 1156.g
• Analytic conductor: $0.577$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{2}$
• CM/RM discs: -4, -68, 17
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1156 = 2^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1156.g (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.576919154604\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{8})\)
Defining polynomial:	\( x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 68)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(i, \sqrt{17})\)
Artin image:	$\OD_{32}$
Artin field:	Galois closure of 16.8.732780301186512843008.1

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\)	\(579\)	\(581\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{8}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

155.1

−0.707107	+	0.707107i
−0.707107	−	0.707107i
0.707107	+	0.707107i
0.707107	−	0.707107i

0.707107

+

0.707107i

0

1.00000i

0

0

0

−0.707107

+

0.707107i

0.707107

−

0.707107i

0

179.1

0.707107

−

0.707107i

0

−

1.00000i

0

0

0

−0.707107

−

0.707107i

0.707107

+

0.707107i

0

399.1

−0.707107

+

0.707107i

0

−

1.00000i

0

0

0

0.707107

+

0.707107i

−0.707107

−

0.707107i

0

423.1

−0.707107

−

0.707107i

0

1.00000i

0

0

0

0.707107

−

0.707107i

−0.707107

+

0.707107i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
17.b	even	2	1	RM by \(\Q(\sqrt{17}) \)
68.d	odd	2	1	CM by \(\Q(\sqrt{-17}) \)
17.c	even	4	2	inner
17.d	even	8	4	inner
68.f	odd	4	2	inner
68.g	odd	8	4	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1156.1.g.a		4
4.b	odd	2	1	CM	1156.1.g.a		4
17.b	even	2	1	RM	1156.1.g.a		4
17.c	even	4	2	inner	1156.1.g.a		4
17.d	even	8	4	inner	1156.1.g.a		4
17.e	odd	16	2		68.1.d.a	✓	1
17.e	odd	16	2		1156.1.c.a		1
17.e	odd	16	4		1156.1.f.a		2
51.i	even	16	2		612.1.e.a		1
68.d	odd	2	1	CM	1156.1.g.a		4
68.f	odd	4	2	inner	1156.1.g.a		4
68.g	odd	8	4	inner	1156.1.g.a		4
68.i	even	16	2		68.1.d.a	✓	1
68.i	even	16	2		1156.1.c.a		1
68.i	even	16	4		1156.1.f.a		2
85.o	even	16	2		1700.1.d.b		2
85.p	odd	16	2		1700.1.h.d		1
85.r	even	16	2		1700.1.d.b		2
119.p	even	16	2		3332.1.g.a		1
119.s	even	48	4		3332.1.o.d		2
119.t	odd	48	4		3332.1.o.c		2
136.q	odd	16	2		1088.1.g.a		1
136.s	even	16	2		1088.1.g.a		1
204.t	odd	16	2		612.1.e.a		1
340.bc	odd	16	2		1700.1.d.b		2
340.bg	even	16	2		1700.1.h.d		1
340.bj	odd	16	2		1700.1.d.b		2
476.bf	odd	16	2		3332.1.g.a		1
476.bk	odd	48	4		3332.1.o.d		2
476.bm	even	48	4		3332.1.o.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
68.1.d.a	✓	1	17.e	odd	16	2
68.1.d.a	✓	1	68.i	even	16	2
612.1.e.a		1	51.i	even	16	2
612.1.e.a		1	204.t	odd	16	2
1088.1.g.a		1	136.q	odd	16	2
1088.1.g.a		1	136.s	even	16	2
1156.1.c.a		1	17.e	odd	16	2
1156.1.c.a		1	68.i	even	16	2
1156.1.f.a		2	17.e	odd	16	4
1156.1.f.a		2	68.i	even	16	4
1156.1.g.a		4	1.a	even	1	1	trivial
1156.1.g.a		4	4.b	odd	2	1	CM
1156.1.g.a		4	17.b	even	2	1	RM
1156.1.g.a		4	17.c	even	4	2	inner
1156.1.g.a		4	17.d	even	8	4	inner
1156.1.g.a		4	68.d	odd	2	1	CM
1156.1.g.a		4	68.f	odd	4	2	inner
1156.1.g.a		4	68.g	odd	8	4	inner
1700.1.d.b		2	85.o	even	16	2
1700.1.d.b		2	85.r	even	16	2
1700.1.d.b		2	340.bc	odd	16	2
1700.1.d.b		2	340.bj	odd	16	2
1700.1.h.d		1	85.p	odd	16	2
1700.1.h.d		1	340.bg	even	16	2
3332.1.g.a		1	119.p	even	16	2
3332.1.g.a		1	476.bf	odd	16	2
3332.1.o.c		2	119.t	odd	48	4
3332.1.o.c		2	476.bm	even	48	4
3332.1.o.d		2	119.s	even	48	4
3332.1.o.d		2	476.bk	odd	48	4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{1}^{\mathrm{new}}(1156, [\chi])\).

Hecke characteristic polynomials

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