Newform orbit 980.2.i.a

• Label: 980.2.i.a
• Level: $980$
• Weight: $2$
• Character orbit: 980.i
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
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 140)
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.

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

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−1.50000

+

2.59808i

0

−0.500000

−

0.866025i

0

0

0

−3.00000

−

5.19615i

0

961.1

0

−1.50000

−

2.59808i

0

−0.500000

+

0.866025i

0

0

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	980.2.i.a		2
7.b	odd	2	1		140.2.i.b	✓	2
7.c	even	3	1		980.2.a.i		1
7.c	even	3	1	inner	980.2.i.a		2
7.d	odd	6	1		140.2.i.b	✓	2
7.d	odd	6	1		980.2.a.a		1
21.c	even	2	1		1260.2.s.b		2
21.g	even	6	1		1260.2.s.b		2
21.g	even	6	1		8820.2.a.w		1
21.h	odd	6	1		8820.2.a.k		1
28.d	even	2	1		560.2.q.a		2
28.f	even	6	1		560.2.q.a		2
28.f	even	6	1		3920.2.a.bi		1
28.g	odd	6	1		3920.2.a.d		1
35.c	odd	2	1		700.2.i.a		2
35.f	even	4	2		700.2.r.c		4
35.i	odd	6	1		700.2.i.a		2
35.i	odd	6	1		4900.2.a.v		1
35.j	even	6	1		4900.2.a.a		1
35.k	even	12	2		700.2.r.c		4
35.k	even	12	2		4900.2.e.c		2
35.l	odd	12	2		4900.2.e.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.i.b	✓	2	7.b	odd	2	1
140.2.i.b	✓	2	7.d	odd	6	1
560.2.q.a		2	28.d	even	2	1
560.2.q.a		2	28.f	even	6	1
700.2.i.a		2	35.c	odd	2	1
700.2.i.a		2	35.i	odd	6	1
700.2.r.c		4	35.f	even	4	2
700.2.r.c		4	35.k	even	12	2
980.2.a.a		1	7.d	odd	6	1
980.2.a.i		1	7.c	even	3	1
980.2.i.a		2	1.a	even	1	1	trivial
980.2.i.a		2	7.c	even	3	1	inner
1260.2.s.b		2	21.c	even	2	1
1260.2.s.b		2	21.g	even	6	1
3920.2.a.d		1	28.g	odd	6	1
3920.2.a.bi		1	28.f	even	6	1
4900.2.a.a		1	35.j	even	6	1
4900.2.a.v		1	35.i	odd	6	1
4900.2.e.b		2	35.l	odd	12	2
4900.2.e.c		2	35.k	even	12	2
8820.2.a.k		1	21.h	odd	6	1
8820.2.a.w		1	21.g	even	6	1

Hecke kernels

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

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

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

Hecke characteristic polynomials

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