Newform orbit 980.2.x.f

• Label: 980.2.x.f
• Level: $980$
• Weight: $2$
• Character orbit: 980.x
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	8.0.12745506816.9
Defining polynomial:	\( x^{8} - 49x^{4} + 2401 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\beta_{6} - \beta_{4} - \beta_{2}) q^{2} + \beta_{5} q^{3} + 2 \beta_{2} q^{4} + (2 \beta_{6} - \beta_{4} - 2 \beta_{2}) q^{5} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{6} + ( - 2 \beta_{6} - 2) q^{8} + (4 \beta_{6} - 4 \beta_{2}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 4 q^{5} - 16 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 49x^{4} + 2401 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 7 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 7 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} ) / 49 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} ) / 49 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} ) / 343 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} ) / 343 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(-\beta_{4}\)	\(-\beta_{6}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

67.1

−0.684771	+	2.55560i
0.684771	−	2.55560i
−2.55560	−	0.684771i
2.55560	+	0.684771i
−2.55560	+	0.684771i
2.55560	−	0.684771i
−0.684771	−	2.55560i
0.684771	+	2.55560i

0.366025

−

1.36603i

−2.55560

+

0.684771i

−1.73205

−

1.00000i

1.23205

−

1.86603i

3.74166i

0

−2.00000

+

2.00000i

3.46410

−

2.00000i

−2.09808

−

2.36603i

67.2

0.366025

−

1.36603i

2.55560

−

0.684771i

−1.73205

−

1.00000i

1.23205

−

1.86603i

−

3.74166i

0

−2.00000

+

2.00000i

3.46410

−

2.00000i

−2.09808

−

2.36603i

263.1

−1.36603

−

0.366025i

−0.684771

−

2.55560i

1.73205

+

1.00000i

−2.23205

+

0.133975i

3.74166i

0

−2.00000

−

2.00000i

−3.46410

+

2.00000i

3.09808

+

0.633975i

263.2

−1.36603

−

0.366025i

0.684771

+

2.55560i

1.73205

+

1.00000i

−2.23205

+

0.133975i

−

3.74166i

0

−2.00000

−

2.00000i

−3.46410

+

2.00000i

3.09808

+

0.633975i

667.1

−1.36603

+

0.366025i

−0.684771

+

2.55560i

1.73205

−

1.00000i

−2.23205

−

0.133975i

−

3.74166i

0

−2.00000

+

2.00000i

−3.46410

−

2.00000i

3.09808

−

0.633975i

667.2

−1.36603

+

0.366025i

0.684771

−

2.55560i

1.73205

−

1.00000i

−2.23205

−

0.133975i

3.74166i

0

−2.00000

+

2.00000i

−3.46410

−

2.00000i

3.09808

−

0.633975i

863.1

0.366025

+

1.36603i

−2.55560

−

0.684771i

−1.73205

+

1.00000i

1.23205

+

1.86603i

−

3.74166i

0

−2.00000

−

2.00000i

3.46410

+

2.00000i

−2.09808

+

2.36603i

863.2

0.366025

+

1.36603i

2.55560

+

0.684771i

−1.73205

+

1.00000i

1.23205

+

1.86603i

3.74166i

0

−2.00000

−

2.00000i

3.46410

+

2.00000i

−2.09808

+

2.36603i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
7.c	even	3	1	inner
20.e	even	4	1	inner
28.g	odd	6	1	inner
35.l	odd	12	1	inner
140.w	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	980.2.x.f		8
4.b	odd	2	1	inner	980.2.x.f		8
5.c	odd	4	1	inner	980.2.x.f		8
7.b	odd	2	1		140.2.w.a	✓	8
7.c	even	3	1		980.2.k.g		4
7.c	even	3	1	inner	980.2.x.f		8
7.d	odd	6	1		140.2.w.a	✓	8
7.d	odd	6	1		980.2.k.e		4
20.e	even	4	1	inner	980.2.x.f		8
28.d	even	2	1		140.2.w.a	✓	8
28.f	even	6	1		140.2.w.a	✓	8
28.f	even	6	1		980.2.k.e		4
28.g	odd	6	1		980.2.k.g		4
28.g	odd	6	1	inner	980.2.x.f		8
35.c	odd	2	1		700.2.be.c		8
35.f	even	4	1		140.2.w.a	✓	8
35.f	even	4	1		700.2.be.c		8
35.i	odd	6	1		700.2.be.c		8
35.k	even	12	1		140.2.w.a	✓	8
35.k	even	12	1		700.2.be.c		8
35.k	even	12	1		980.2.k.e		4
35.l	odd	12	1		980.2.k.g		4
35.l	odd	12	1	inner	980.2.x.f		8
140.c	even	2	1		700.2.be.c		8
140.j	odd	4	1		140.2.w.a	✓	8
140.j	odd	4	1		700.2.be.c		8
140.s	even	6	1		700.2.be.c		8
140.w	even	12	1		980.2.k.g		4
140.w	even	12	1	inner	980.2.x.f		8
140.x	odd	12	1		140.2.w.a	✓	8
140.x	odd	12	1		700.2.be.c		8
140.x	odd	12	1		980.2.k.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.w.a	✓	8	7.b	odd	2	1
140.2.w.a	✓	8	7.d	odd	6	1
140.2.w.a	✓	8	28.d	even	2	1
140.2.w.a	✓	8	28.f	even	6	1
140.2.w.a	✓	8	35.f	even	4	1
140.2.w.a	✓	8	35.k	even	12	1
140.2.w.a	✓	8	140.j	odd	4	1
140.2.w.a	✓	8	140.x	odd	12	1
700.2.be.c		8	35.c	odd	2	1
700.2.be.c		8	35.f	even	4	1
700.2.be.c		8	35.i	odd	6	1
700.2.be.c		8	35.k	even	12	1
700.2.be.c		8	140.c	even	2	1
700.2.be.c		8	140.j	odd	4	1
700.2.be.c		8	140.s	even	6	1
700.2.be.c		8	140.x	odd	12	1
980.2.k.e		4	7.d	odd	6	1
980.2.k.e		4	28.f	even	6	1
980.2.k.e		4	35.k	even	12	1
980.2.k.e		4	140.x	odd	12	1
980.2.k.g		4	7.c	even	3	1
980.2.k.g		4	28.g	odd	6	1
980.2.k.g		4	35.l	odd	12	1
980.2.k.g		4	140.w	even	12	1
980.2.x.f		8	1.a	even	1	1	trivial
980.2.x.f		8	4.b	odd	2	1	inner
980.2.x.f		8	5.c	odd	4	1	inner
980.2.x.f		8	7.c	even	3	1	inner
980.2.x.f		8	20.e	even	4	1	inner
980.2.x.f		8	28.g	odd	6	1	inner
980.2.x.f		8	35.l	odd	12	1	inner
980.2.x.f		8	140.w	even	12	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{8} - 49T_{3}^{4} + 2401 \)

\( T_{11}^{4} - 14T_{11}^{2} + 196 \)

\( T_{13}^{2} - 4T_{13} + 8 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^4 + 2*T^3 + 2*T^2 + 4*T + 4)^2
$3$	\( T^{8} - 49T^{4} + 2401 \)
$5$	(T^4 + 2*T^3 - T^2 + 10*T + 25)^2
$7$	\( T^{8} \)
$11$	\( (T^{4} - 14 T^{2} + 196)^{2} \)