Newform orbit 1156.1.f.a

• Label: 1156.1.f.a
• Level: $1156$
• Weight: $1$
• Character orbit: 1156.f
• Analytic conductor: $0.577$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -4, -68, 17
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.576919154604\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
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:	$\OD_{16}$
Artin field:	Galois closure of 8.4.6565418768.1

$q$-expansion

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

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

Character values

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

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

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

251.1

	−	1.00000i
		1.00000i

−

1.00000i

0

−1.00000

0

0

0

1.00000i

1.00000i

0

327.1

1.00000i

0

−1.00000

0

0

0

−

1.00000i

−

1.00000i

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
68.f	odd	4	2	inner

Twists

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

    

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

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