Newform orbit 1156.2.e.b

• Label: 1156.2.e.b
• Level: $1156$
• Weight: $2$
• Character orbit: 1156.e
• Analytic conductor: $9.231$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.23070647366\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 68)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (i + 1) q^{3} + ( - 2 i - 2) q^{5} + (3 i - 3) q^{7} - i q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} - 6 q^{7}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

829.1

		1.00000i
	−	1.00000i

0

1.00000

+

1.00000i

0

−2.00000

−

2.00000i

0

−3.00000

+

3.00000i

0

−

1.00000i

0

905.1

0

1.00000

−

1.00000i

0

−2.00000

+

2.00000i

0

−3.00000

−

3.00000i

0

1.00000i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.c	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1156.2.e.b		2
17.b	even	2	1		1156.2.e.a		2
17.c	even	4	1		1156.2.e.a		2
17.c	even	4	1	inner	1156.2.e.b		2
17.d	even	8	2		68.2.b.a	✓	2
17.d	even	8	2		1156.2.a.c		2
17.e	odd	16	8		1156.2.h.d		8
51.g	odd	8	2		612.2.b.a		2
68.g	odd	8	2		272.2.b.c		2
68.g	odd	8	2		4624.2.a.n		2
85.k	odd	8	2		1700.2.g.a		4
85.m	even	8	2		1700.2.c.a		2
85.n	odd	8	2		1700.2.g.a		4
119.l	odd	8	2		3332.2.b.a		2
136.o	even	8	2		1088.2.b.e		2
136.p	odd	8	2		1088.2.b.f		2
204.p	even	8	2		2448.2.c.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
68.2.b.a	✓	2	17.d	even	8	2
272.2.b.c		2	68.g	odd	8	2
612.2.b.a		2	51.g	odd	8	2
1088.2.b.e		2	136.o	even	8	2
1088.2.b.f		2	136.p	odd	8	2
1156.2.a.c		2	17.d	even	8	2
1156.2.e.a		2	17.b	even	2	1
1156.2.e.a		2	17.c	even	4	1
1156.2.e.b		2	1.a	even	1	1	trivial
1156.2.e.b		2	17.c	even	4	1	inner
1156.2.h.d		8	17.e	odd	16	8
1700.2.c.a		2	85.m	even	8	2
1700.2.g.a		4	85.k	odd	8	2
1700.2.g.a		4	85.n	odd	8	2
2448.2.c.d		2	204.p	even	8	2
3332.2.b.a		2	119.l	odd	8	2
4624.2.a.n		2	68.g	odd	8	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1156, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2T + 2 \)
$5$	\( T^{2} + 4T + 8 \)
$7$	\( T^{2} + 6T + 18 \)
$11$	\( T^{2} - 2T + 2 \)