Newform orbit 931.2.f.b

• Label: 931.2.f.b
• Level: $931$
• Weight: $2$
• Character orbit: 931.f
• Analytic conductor: $7.434$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 931 = 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	931.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.43407242818\)
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 19)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/931\mathbb{Z}\right)^\times\).

\(n\)	\(248\)	\(344\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

324.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−1.00000

−

1.73205i

1.00000

+

1.73205i

1.50000

−

2.59808i

0

0

0

−0.500000

+

0.866025i

0

704.1

0

−1.00000

+

1.73205i

1.00000

−

1.73205i

1.50000

+

2.59808i

0

0

0

−0.500000

−

0.866025i

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	931.2.f.b		2
7.b	odd	2	1		931.2.f.c		2
7.c	even	3	1		931.2.a.a		1
7.c	even	3	1	inner	931.2.f.b		2
7.d	odd	6	1		19.2.a.a	✓	1
7.d	odd	6	1		931.2.f.c		2
21.g	even	6	1		171.2.a.b		1
21.h	odd	6	1		8379.2.a.j		1
28.f	even	6	1		304.2.a.f		1
35.i	odd	6	1		475.2.a.b		1
35.k	even	12	2		475.2.b.a		2
56.j	odd	6	1		1216.2.a.o		1
56.m	even	6	1		1216.2.a.b		1
77.i	even	6	1		2299.2.a.b		1
84.j	odd	6	1		2736.2.a.c		1
91.s	odd	6	1		3211.2.a.a		1
105.p	even	6	1		4275.2.a.i		1
119.h	odd	6	1		5491.2.a.b		1
133.i	even	6	1		361.2.c.a		2
133.k	odd	6	1		361.2.c.c		2
133.o	even	6	1		361.2.a.b		1
133.s	even	6	1		361.2.c.a		2
133.t	odd	6	1		361.2.c.c		2
133.x	odd	18	3		361.2.e.d		6
133.z	odd	18	3		361.2.e.d		6
133.bb	even	18	3		361.2.e.e		6
133.bf	even	18	3		361.2.e.e		6
140.s	even	6	1		7600.2.a.c		1
399.s	odd	6	1		3249.2.a.d		1
532.bh	odd	6	1		5776.2.a.c		1
665.y	even	6	1		9025.2.a.d		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.2.a.a	✓	1	7.d	odd	6	1
171.2.a.b		1	21.g	even	6	1
304.2.a.f		1	28.f	even	6	1
361.2.a.b		1	133.o	even	6	1
361.2.c.a		2	133.i	even	6	1
361.2.c.a		2	133.s	even	6	1
361.2.c.c		2	133.k	odd	6	1
361.2.c.c		2	133.t	odd	6	1
361.2.e.d		6	133.x	odd	18	3
361.2.e.d		6	133.z	odd	18	3
361.2.e.e		6	133.bb	even	18	3
361.2.e.e		6	133.bf	even	18	3
475.2.a.b		1	35.i	odd	6	1
475.2.b.a		2	35.k	even	12	2
931.2.a.a		1	7.c	even	3	1
931.2.f.b		2	1.a	even	1	1	trivial
931.2.f.b		2	7.c	even	3	1	inner
931.2.f.c		2	7.b	odd	2	1
931.2.f.c		2	7.d	odd	6	1
1216.2.a.b		1	56.m	even	6	1
1216.2.a.o		1	56.j	odd	6	1
2299.2.a.b		1	77.i	even	6	1
2736.2.a.c		1	84.j	odd	6	1
3211.2.a.a		1	91.s	odd	6	1
3249.2.a.d		1	399.s	odd	6	1
4275.2.a.i		1	105.p	even	6	1
5491.2.a.b		1	119.h	odd	6	1
5776.2.a.c		1	532.bh	odd	6	1
7600.2.a.c		1	140.s	even	6	1
8379.2.a.j		1	21.h	odd	6	1
9025.2.a.d		1	665.y	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2} \)

\( T_{3}^{2} + 2T_{3} + 4 \)

Hecke characteristic polynomials

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