Newform orbit 980.2.k.h

• Label: 980.2.k.h
• Level: $980$
• Weight: $2$
• Character orbit: 980.k
• 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.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( - \beta_{6} + \beta_{3}) q^{2} + ( - \beta_{7} - \beta_{4}) q^{3} + ( - \beta_{6} - \beta_{2}) q^{4} + ( - 2 \beta_{5} + \beta_1) q^{5} + (2 \beta_{5} - \beta_{4} + 2 \beta_1) q^{6} + ( - 2 \beta_{3} + 2) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 16 q^{8}+O(q^{10}) \)

Basis of coefficient ring

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

Character values

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

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(1\)	\(\beta_{3}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

687.1

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

−0.366025

+

1.36603i

−1.22474

−

1.22474i

−1.73205

−

1.00000i

−0.707107

+

2.12132i

2.12132

−

1.22474i

0

2.00000

−

2.00000i

0

−2.63896

−

1.74238i

687.2

−0.366025

+

1.36603i

1.22474

+

1.22474i

−1.73205

−

1.00000i

0.707107

−

2.12132i

−2.12132

+

1.22474i

0

2.00000

−

2.00000i

0

2.63896

+

1.74238i

687.3

1.36603

−

0.366025i

−1.22474

−

1.22474i

1.73205

−

1.00000i

0.707107

−

2.12132i

−2.12132

−

1.22474i

0

2.00000

−

2.00000i

0

0.189469

−

3.15660i

687.4

1.36603

−

0.366025i

1.22474

+

1.22474i

1.73205

−

1.00000i

−0.707107

+

2.12132i

2.12132

+

1.22474i

0

2.00000

−

2.00000i

0

−0.189469

+

3.15660i

883.1

−0.366025

−

1.36603i

−1.22474

+

1.22474i

−1.73205

+

1.00000i

−0.707107

−

2.12132i

2.12132

+

1.22474i

0

2.00000

+

2.00000i

0

−2.63896

+

1.74238i

883.2

−0.366025

−

1.36603i

1.22474

−

1.22474i

−1.73205

+

1.00000i

0.707107

+

2.12132i

−2.12132

−

1.22474i

0

2.00000

+

2.00000i

0

2.63896

−

1.74238i

883.3

1.36603

+

0.366025i

−1.22474

+

1.22474i

1.73205

+

1.00000i

0.707107

+

2.12132i

−2.12132

+

1.22474i

0

2.00000

+

2.00000i

0

0.189469

+

3.15660i

883.4

1.36603

+

0.366025i

1.22474

−

1.22474i

1.73205

+

1.00000i

−0.707107

−

2.12132i

2.12132

−

1.22474i

0

2.00000

+

2.00000i

0

−0.189469

−

3.15660i

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.b	odd	2	1	inner
20.e	even	4	1	inner
28.d	even	2	1	inner
35.f	even	4	1	inner
140.j	odd	4	1	inner

Twists

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

    

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

Hecke kernels

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

\( T_{3}^{4} + 9 \)

\( T_{13}^{4} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^4 - 2*T^3 + 2*T^2 - 4*T + 4)^2
$3$	\( (T^{4} + 9)^{2} \)
$5$	\( (T^{4} + 8 T^{2} + 25)^{2} \)
$7$	\( T^{8} \)
$11$	\( (T^{2} + 3)^{4} \)