Newform orbit 980.1.f.b

• Label: 980.1.f.b
• Level: $980$
• Weight: $1$
• Character orbit: 980.f
• Self dual: yes
• Analytic conductor: $0.489$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -20
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	980.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.489083712380\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 140)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.980.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.3841600.1
Stark unit:	Root of $x^{6} - 167x^{5} - 272x^{4} - 28769x^{3} - 272x^{2} - 167x + 1$

$q$-expansion

\(f(q)\)

\(=\)

q - q^2 + q^3 + q^4 + q^5 - q^6 - q^8 - q^10 + q^12 + q^15 + q^16 + q^20 + q^23 - q^24 + q^25 - q^27 - q^29 - q^30 - q^32 - q^40 - q^41 + q^43 - q^46 - 2 * q^47 + q^48 - q^50 + q^54 + q^58 + q^60 - q^61 + q^64 + q^67 + q^69 + q^75 + q^80 - q^81 + q^82 + q^83 - q^86 - q^87 - q^89 + q^92 + 2 * q^94 - q^96

Character values

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

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(0\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

99.1

0

−1.00000

1.00000

1.00000

1.00000

−1.00000

0

−1.00000

0

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	980.1.f.b		1
4.b	odd	2	1		980.1.f.c		1
5.b	even	2	1		980.1.f.c		1
7.b	odd	2	1		980.1.f.a		1
7.c	even	3	2		140.1.p.b	yes	2
7.d	odd	6	2		980.1.p.b		2
20.d	odd	2	1	CM	980.1.f.b		1
21.h	odd	6	2		1260.1.ci.a		2
28.d	even	2	1		980.1.f.d		1
28.f	even	6	2		980.1.p.a		2
28.g	odd	6	2		140.1.p.a	✓	2
35.c	odd	2	1		980.1.f.d		1
35.i	odd	6	2		980.1.p.a		2
35.j	even	6	2		140.1.p.a	✓	2
35.l	odd	12	4		700.1.u.a		4
56.k	odd	6	2		2240.1.bt.a		2
56.p	even	6	2		2240.1.bt.b		2
84.n	even	6	2		1260.1.ci.b		2
105.o	odd	6	2		1260.1.ci.b		2
140.c	even	2	1		980.1.f.a		1
140.p	odd	6	2		140.1.p.b	yes	2
140.s	even	6	2		980.1.p.b		2
140.w	even	12	4		700.1.u.a		4
280.bf	even	6	2		2240.1.bt.a		2
280.bi	odd	6	2		2240.1.bt.b		2
420.ba	even	6	2		1260.1.ci.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.1.p.a	✓	2	28.g	odd	6	2
140.1.p.a	✓	2	35.j	even	6	2
140.1.p.b	yes	2	7.c	even	3	2
140.1.p.b	yes	2	140.p	odd	6	2
700.1.u.a		4	35.l	odd	12	4
700.1.u.a		4	140.w	even	12	4
980.1.f.a		1	7.b	odd	2	1
980.1.f.a		1	140.c	even	2	1
980.1.f.b		1	1.a	even	1	1	trivial
980.1.f.b		1	20.d	odd	2	1	CM
980.1.f.c		1	4.b	odd	2	1
980.1.f.c		1	5.b	even	2	1
980.1.f.d		1	28.d	even	2	1
980.1.f.d		1	35.c	odd	2	1
980.1.p.a		2	28.f	even	6	2
980.1.p.a		2	35.i	odd	6	2
980.1.p.b		2	7.d	odd	6	2
980.1.p.b		2	140.s	even	6	2
1260.1.ci.a		2	21.h	odd	6	2
1260.1.ci.a		2	420.ba	even	6	2
1260.1.ci.b		2	84.n	even	6	2
1260.1.ci.b		2	105.o	odd	6	2
2240.1.bt.a		2	56.k	odd	6	2
2240.1.bt.a		2	280.bf	even	6	2
2240.1.bt.b		2	56.p	even	6	2
2240.1.bt.b		2	280.bi	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(980, [\chi])\):

\( T_{3} - 1 \)

\( T_{23} - 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 1 \)
$3$	\( T - 1 \)
$5$	\( T - 1 \)
$7$	\( T \)
$11$	\( T \)