Newform orbit 560.2.g.c

• Label: 560.2.g.c
• Level: $560$
• Weight: $2$
• Character orbit: 560.g
• Analytic conductor: $4.472$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47162251319\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(337\)	\(351\)	\(421\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

449.1

	−	1.00000i
		1.00000i

0

0

0

1.00000

−

2.00000i

0

1.00000i

0

3.00000

0

449.2

0

0

0

1.00000

+

2.00000i

0

−

1.00000i

0

3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	560.2.g.c		2
3.b	odd	2	1		5040.2.t.g		2
4.b	odd	2	1		140.2.e.b	✓	2
5.b	even	2	1	inner	560.2.g.c		2
5.c	odd	4	1		2800.2.a.o		1
5.c	odd	4	1		2800.2.a.s		1
8.b	even	2	1		2240.2.g.c		2
8.d	odd	2	1		2240.2.g.d		2
12.b	even	2	1		1260.2.k.b		2
15.d	odd	2	1		5040.2.t.g		2
20.d	odd	2	1		140.2.e.b	✓	2
20.e	even	4	1		700.2.a.f		1
20.e	even	4	1		700.2.a.h		1
28.d	even	2	1		980.2.e.a		2
28.f	even	6	2		980.2.q.e		4
28.g	odd	6	2		980.2.q.d		4
40.e	odd	2	1		2240.2.g.d		2
40.f	even	2	1		2240.2.g.c		2
60.h	even	2	1		1260.2.k.b		2
60.l	odd	4	1		6300.2.a.g		1
60.l	odd	4	1		6300.2.a.y		1
140.c	even	2	1		980.2.e.a		2
140.j	odd	4	1		4900.2.a.l		1
140.j	odd	4	1		4900.2.a.m		1
140.p	odd	6	2		980.2.q.d		4
140.s	even	6	2		980.2.q.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.e.b	✓	2	4.b	odd	2	1
140.2.e.b	✓	2	20.d	odd	2	1
560.2.g.c		2	1.a	even	1	1	trivial
560.2.g.c		2	5.b	even	2	1	inner
700.2.a.f		1	20.e	even	4	1
700.2.a.h		1	20.e	even	4	1
980.2.e.a		2	28.d	even	2	1
980.2.e.a		2	140.c	even	2	1
980.2.q.d		4	28.g	odd	6	2
980.2.q.d		4	140.p	odd	6	2
980.2.q.e		4	28.f	even	6	2
980.2.q.e		4	140.s	even	6	2
1260.2.k.b		2	12.b	even	2	1
1260.2.k.b		2	60.h	even	2	1
2240.2.g.c		2	8.b	even	2	1
2240.2.g.c		2	40.f	even	2	1
2240.2.g.d		2	8.d	odd	2	1
2240.2.g.d		2	40.e	odd	2	1
2800.2.a.o		1	5.c	odd	4	1
2800.2.a.s		1	5.c	odd	4	1
4900.2.a.l		1	140.j	odd	4	1
4900.2.a.m		1	140.j	odd	4	1
5040.2.t.g		2	3.b	odd	2	1
5040.2.t.g		2	15.d	odd	2	1
6300.2.a.g		1	60.l	odd	4	1
6300.2.a.y		1	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\):

\( T_{3} \)

\( T_{11} \)

Hecke characteristic polynomials

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