Newform orbit 490.2.i.f

• Label: 490.2.i.f
• Level: $490$
• Weight: $2$
• Character orbit: 490.i
• Analytic conductor: $3.913$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	490.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.91266969904\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{6} - \beta_{5}) q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{4} + \beta_{2} - \beta_1) q^{5} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{6} + \beta_{4} q^{8} + 3 \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} + 4 q^{5} + 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{24}^{2} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{24}^{4} \)
\(\beta_{3}\)	\(=\)	\( \zeta_{24}^{5} + \zeta_{24} \)
\(\beta_{4}\)	\(=\)	\( \zeta_{24}^{6} \)
\(\beta_{5}\)	\(=\)	\( \zeta_{24}^{7} + \zeta_{24}^{3} \)
\(\beta_{6}\)	\(=\)	\( -\zeta_{24}^{5} + 2\zeta_{24} \)
\(\beta_{7}\)	\(=\)	\( -\zeta_{24}^{7} + 2\zeta_{24}^{3} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(-\beta_{2}\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

79.1

0.965926	−	0.258819i
−0.965926	+	0.258819i
−0.258819	−	0.965926i
0.258819	+	0.965926i
0.965926	+	0.258819i
−0.965926	−	0.258819i
−0.258819	+	0.965926i
0.258819	−	0.965926i

−0.866025

−

0.500000i

−2.12132

+

1.22474i

0.500000

+

0.866025i

−2.03906

−

0.917738i

2.44949

0

−

1.00000i

1.50000

−

2.59808i

1.30701

+

1.81431i

79.2

−0.866025

−

0.500000i

2.12132

−

1.22474i

0.500000

+

0.866025i

1.30701

−

1.81431i

−2.44949

0

−

1.00000i

1.50000

−

2.59808i

−2.03906

+

0.917738i

79.3

0.866025

+

0.500000i

−2.12132

+

1.22474i

0.500000

+

0.866025i

0.917738

−

2.03906i

−2.44949

0

1.00000i

1.50000

−

2.59808i

1.81431

−

1.30701i

79.4

0.866025

+

0.500000i

2.12132

−

1.22474i

0.500000

+

0.866025i

1.81431

+

1.30701i

2.44949

0

1.00000i

1.50000

−

2.59808i

0.917738

+

2.03906i

459.1

−0.866025

+

0.500000i

−2.12132

−

1.22474i

0.500000

−

0.866025i

−2.03906

+

0.917738i

2.44949

0

1.00000i

1.50000

+

2.59808i

1.30701

−

1.81431i

459.2

−0.866025

+

0.500000i

2.12132

+

1.22474i

0.500000

−

0.866025i

1.30701

+

1.81431i

−2.44949

0

1.00000i

1.50000

+

2.59808i

−2.03906

−

0.917738i

459.3

0.866025

−

0.500000i

−2.12132

−

1.22474i

0.500000

−

0.866025i

0.917738

+

2.03906i

−2.44949

0

−

1.00000i

1.50000

+

2.59808i

1.81431

+

1.30701i

459.4

0.866025

−

0.500000i

2.12132

+

1.22474i

0.500000

−

0.866025i

1.81431

−

1.30701i

2.44949

0

−

1.00000i

1.50000

+

2.59808i

0.917738

−

2.03906i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.c	even	3	1	inner
35.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	490.2.i.f		8
5.b	even	2	1	inner	490.2.i.f		8
7.b	odd	2	1		490.2.i.c		8
7.c	even	3	1		490.2.c.e		4
7.c	even	3	1	inner	490.2.i.f		8
7.d	odd	6	1		70.2.c.a	✓	4
7.d	odd	6	1		490.2.i.c		8
21.g	even	6	1		630.2.g.g		4
28.f	even	6	1		560.2.g.e		4
35.c	odd	2	1		490.2.i.c		8
35.i	odd	6	1		70.2.c.a	✓	4
35.i	odd	6	1		490.2.i.c		8
35.j	even	6	1		490.2.c.e		4
35.j	even	6	1	inner	490.2.i.f		8
35.k	even	12	1		350.2.a.g		2
35.k	even	12	1		350.2.a.h		2
35.l	odd	12	1		2450.2.a.bl		2
35.l	odd	12	1		2450.2.a.bq		2
56.j	odd	6	1		2240.2.g.j		4
56.m	even	6	1		2240.2.g.i		4
84.j	odd	6	1		5040.2.t.t		4
105.p	even	6	1		630.2.g.g		4
105.w	odd	12	1		3150.2.a.bs		2
105.w	odd	12	1		3150.2.a.bt		2
140.s	even	6	1		560.2.g.e		4
140.x	odd	12	1		2800.2.a.bl		2
140.x	odd	12	1		2800.2.a.bm		2
280.ba	even	6	1		2240.2.g.i		4
280.bk	odd	6	1		2240.2.g.j		4
420.be	odd	6	1		5040.2.t.t		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.2.c.a	✓	4	7.d	odd	6	1
70.2.c.a	✓	4	35.i	odd	6	1
350.2.a.g		2	35.k	even	12	1
350.2.a.h		2	35.k	even	12	1
490.2.c.e		4	7.c	even	3	1
490.2.c.e		4	35.j	even	6	1
490.2.i.c		8	7.b	odd	2	1
490.2.i.c		8	7.d	odd	6	1
490.2.i.c		8	35.c	odd	2	1
490.2.i.c		8	35.i	odd	6	1
490.2.i.f		8	1.a	even	1	1	trivial
490.2.i.f		8	5.b	even	2	1	inner
490.2.i.f		8	7.c	even	3	1	inner
490.2.i.f		8	35.j	even	6	1	inner
560.2.g.e		4	28.f	even	6	1
560.2.g.e		4	140.s	even	6	1
630.2.g.g		4	21.g	even	6	1
630.2.g.g		4	105.p	even	6	1
2240.2.g.i		4	56.m	even	6	1
2240.2.g.i		4	280.ba	even	6	1
2240.2.g.j		4	56.j	odd	6	1
2240.2.g.j		4	280.bk	odd	6	1
2450.2.a.bl		2	35.l	odd	12	1
2450.2.a.bq		2	35.l	odd	12	1
2800.2.a.bl		2	140.x	odd	12	1
2800.2.a.bm		2	140.x	odd	12	1
3150.2.a.bs		2	105.w	odd	12	1
3150.2.a.bt		2	105.w	odd	12	1
5040.2.t.t		4	84.j	odd	6	1
5040.2.t.t		4	420.be	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{4} - 6T_{3}^{2} + 36 \)

\( T_{11}^{4} + 24T_{11}^{2} + 576 \)

\( T_{19}^{4} + 8T_{19}^{3} + 54T_{19}^{2} + 80T_{19} + 100 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{2} + 1)^{2} \)
$3$	\( (T^{4} - 6 T^{2} + 36)^{2} \)
$5$	T^8 - 4*T^7 + 8*T^6 + 8*T^5 - 41*T^4 + 40*T^3 + 200*T^2 - 500*T + 625
$7$	\( T^{8} \)
$11$	\( (T^{4} + 24 T^{2} + 576)^{2} \)