Newform orbit 1156.1.c.b

• Label: 1156.1.c.b
• Level: $1156$
• Weight: $1$
• Character orbit: 1156.c
• Self dual: yes
• Analytic conductor: $0.577$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1156 = 2^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1156.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.576919154604\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 68)
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.2.19652.1
Artin image:	$D_8$
Artin field:	Galois closure of 8.2.26261675072.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{2} + q^{4} - \beta q^{5} - q^{8} + q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(579\)	\(581\)
\(\chi(n)\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

579.1

1.41421
−1.41421

−1.00000

0

1.00000

−1.41421

0

0

−1.00000

1.00000

1.41421

579.2

−1.00000

0

1.00000

1.41421

0

0

−1.00000

1.00000

−1.41421

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	inner
68.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1156.1.c.b		2
4.b	odd	2	1	CM	1156.1.c.b		2
17.b	even	2	1	inner	1156.1.c.b		2
17.c	even	4	2		1156.1.d.a		2
17.d	even	8	2		68.1.f.a	✓	2
17.d	even	8	2		1156.1.f.b		2
17.e	odd	16	8		1156.1.g.b		8
51.g	odd	8	2		612.1.l.a		2
68.d	odd	2	1	inner	1156.1.c.b		2
68.f	odd	4	2		1156.1.d.a		2
68.g	odd	8	2		68.1.f.a	✓	2
68.g	odd	8	2		1156.1.f.b		2
68.i	even	16	8		1156.1.g.b		8
85.k	odd	8	2		1700.1.n.b		2
85.m	even	8	2		1700.1.p.a		2
85.n	odd	8	2		1700.1.n.a		2
119.l	odd	8	2		3332.1.m.b		2
119.q	even	24	4		3332.1.bc.c		4
119.r	odd	24	4		3332.1.bc.b		4
136.o	even	8	2		1088.1.p.a		2
136.p	odd	8	2		1088.1.p.a		2
204.p	even	8	2		612.1.l.a		2
340.w	even	8	2		1700.1.n.a		2
340.z	even	8	2		1700.1.n.b		2
340.ba	odd	8	2		1700.1.p.a		2
476.w	even	8	2		3332.1.m.b		2
476.bg	odd	24	4		3332.1.bc.c		4
476.bj	even	24	4		3332.1.bc.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
68.1.f.a	✓	2	17.d	even	8	2
68.1.f.a	✓	2	68.g	odd	8	2
612.1.l.a		2	51.g	odd	8	2
612.1.l.a		2	204.p	even	8	2
1088.1.p.a		2	136.o	even	8	2
1088.1.p.a		2	136.p	odd	8	2
1156.1.c.b		2	1.a	even	1	1	trivial
1156.1.c.b		2	4.b	odd	2	1	CM
1156.1.c.b		2	17.b	even	2	1	inner
1156.1.c.b		2	68.d	odd	2	1	inner
1156.1.d.a		2	17.c	even	4	2
1156.1.d.a		2	68.f	odd	4	2
1156.1.f.b		2	17.d	even	8	2
1156.1.f.b		2	68.g	odd	8	2
1156.1.g.b		8	17.e	odd	16	8
1156.1.g.b		8	68.i	even	16	8
1700.1.n.a		2	85.n	odd	8	2
1700.1.n.a		2	340.w	even	8	2
1700.1.n.b		2	85.k	odd	8	2
1700.1.n.b		2	340.z	even	8	2
1700.1.p.a		2	85.m	even	8	2
1700.1.p.a		2	340.ba	odd	8	2
3332.1.m.b		2	119.l	odd	8	2
3332.1.m.b		2	476.w	even	8	2
3332.1.bc.b		4	119.r	odd	24	4
3332.1.bc.b		4	476.bj	even	24	4
3332.1.bc.c		4	119.q	even	24	4
3332.1.bc.c		4	476.bg	odd	24	4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2 \) acting on \(S_{1}^{\mathrm{new}}(1156, [\chi])\).

Hecke characteristic polynomials

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