Newform orbit 560.2.q.i

• Label: 560.2.q.i
• Level: $560$
• Weight: $2$
• Character orbit: 560.q
• 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.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47162251319\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\)	\(241\)	\(337\)	\(351\)	\(421\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

81.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

1.50000

−

2.59808i

0

0.500000

+

0.866025i

0

−0.500000

−

2.59808i

0

−3.00000

−

5.19615i

0

401.1

0

1.50000

+

2.59808i

0

0.500000

−

0.866025i

0

−0.500000

+

2.59808i

0

−3.00000

+

5.19615i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	560.2.q.i		2
4.b	odd	2	1		70.2.e.a	✓	2
7.c	even	3	1	inner	560.2.q.i		2
7.c	even	3	1		3920.2.a.b		1
7.d	odd	6	1		3920.2.a.bk		1
12.b	even	2	1		630.2.k.f		2
20.d	odd	2	1		350.2.e.l		2
20.e	even	4	2		350.2.j.f		4
28.d	even	2	1		490.2.e.f		2
28.f	even	6	1		490.2.a.e		1
28.f	even	6	1		490.2.e.f		2
28.g	odd	6	1		70.2.e.a	✓	2
28.g	odd	6	1		490.2.a.k		1
84.j	odd	6	1		4410.2.a.h		1
84.n	even	6	1		630.2.k.f		2
84.n	even	6	1		4410.2.a.r		1
140.p	odd	6	1		350.2.e.l		2
140.p	odd	6	1		2450.2.a.b		1
140.s	even	6	1		2450.2.a.q		1
140.w	even	12	2		350.2.j.f		4
140.w	even	12	2		2450.2.c.s		2
140.x	odd	12	2		2450.2.c.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.2.e.a	✓	2	4.b	odd	2	1
70.2.e.a	✓	2	28.g	odd	6	1
350.2.e.l		2	20.d	odd	2	1
350.2.e.l		2	140.p	odd	6	1
350.2.j.f		4	20.e	even	4	2
350.2.j.f		4	140.w	even	12	2
490.2.a.e		1	28.f	even	6	1
490.2.a.k		1	28.g	odd	6	1
490.2.e.f		2	28.d	even	2	1
490.2.e.f		2	28.f	even	6	1
560.2.q.i		2	1.a	even	1	1	trivial
560.2.q.i		2	7.c	even	3	1	inner
630.2.k.f		2	12.b	even	2	1
630.2.k.f		2	84.n	even	6	1
2450.2.a.b		1	140.p	odd	6	1
2450.2.a.q		1	140.s	even	6	1
2450.2.c.a		2	140.x	odd	12	2
2450.2.c.s		2	140.w	even	12	2
3920.2.a.b		1	7.c	even	3	1
3920.2.a.bk		1	7.d	odd	6	1
4410.2.a.h		1	84.j	odd	6	1
4410.2.a.r		1	84.n	even	6	1

Hecke kernels

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

\( T_{3}^{2} - 3T_{3} + 9 \)

\( T_{11}^{2} + 2T_{11} + 4 \)

\( T_{13} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 3T + 9 \)
$5$	\( T^{2} - T + 1 \)
$7$	\( T^{2} + T + 7 \)
$11$	\( T^{2} + 2T + 4 \)