Newform orbit 931.2.o.b

• Label: 931.2.o.b
• Level: $931$
• Weight: $2$
• Character orbit: 931.o
• Analytic conductor: $7.434$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -19
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 931 = 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	931.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.43407242818\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial:	\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 133)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{3} + 4\nu^{2} + 36\nu - 25 ) / 20 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20 \)
\(\beta_{3}\)	\(=\)	\( ( -9\nu^{3} + 4\nu^{2} + 36\nu + 45 ) / 20 \)

Character values

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

\(n\)	\(248\)	\(344\)
\(\chi(n)\)	\(1 - \beta_{2}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

227.1

2.13746	−	0.656712i
−1.63746	+	1.52274i
2.13746	+	0.656712i
−1.63746	−	1.52274i

0

0

−1.00000

+

1.73205i

−3.77492

+

2.17945i

0

0

0

1.50000

+

2.59808i

0

227.2

0

0

−1.00000

+

1.73205i

3.77492

−

2.17945i

0

0

0

1.50000

+

2.59808i

0

607.1

0

0

−1.00000

−

1.73205i

−3.77492

−

2.17945i

0

0

0

1.50000

−

2.59808i

0

607.2

0

0

−1.00000

−

1.73205i

3.77492

+

2.17945i

0

0

0

1.50000

−

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by \(\Q(\sqrt{-19}) \)
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
133.c	even	2	1	inner
133.o	even	6	1	inner
133.r	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	931.2.o.b		4
7.b	odd	2	1	inner	931.2.o.b		4
7.c	even	3	1		133.2.c.a	✓	2
7.c	even	3	1	inner	931.2.o.b		4
7.d	odd	6	1		133.2.c.a	✓	2
7.d	odd	6	1	inner	931.2.o.b		4
19.b	odd	2	1	CM	931.2.o.b		4
21.g	even	6	1		1197.2.c.c		2
21.h	odd	6	1		1197.2.c.c		2
28.f	even	6	1		2128.2.m.a		2
28.g	odd	6	1		2128.2.m.a		2
133.c	even	2	1	inner	931.2.o.b		4
133.o	even	6	1		133.2.c.a	✓	2
133.o	even	6	1	inner	931.2.o.b		4
133.r	odd	6	1		133.2.c.a	✓	2
133.r	odd	6	1	inner	931.2.o.b		4
399.s	odd	6	1		1197.2.c.c		2
399.w	even	6	1		1197.2.c.c		2
532.t	even	6	1		2128.2.m.a		2
532.bh	odd	6	1		2128.2.m.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
133.2.c.a	✓	2	7.c	even	3	1
133.2.c.a	✓	2	7.d	odd	6	1
133.2.c.a	✓	2	133.o	even	6	1
133.2.c.a	✓	2	133.r	odd	6	1
931.2.o.b		4	1.a	even	1	1	trivial
931.2.o.b		4	7.b	odd	2	1	inner
931.2.o.b		4	7.c	even	3	1	inner
931.2.o.b		4	7.d	odd	6	1	inner
931.2.o.b		4	19.b	odd	2	1	CM
931.2.o.b		4	133.c	even	2	1	inner
931.2.o.b		4	133.o	even	6	1	inner
931.2.o.b		4	133.r	odd	6	1	inner
1197.2.c.c		2	21.g	even	6	1
1197.2.c.c		2	21.h	odd	6	1
1197.2.c.c		2	399.s	odd	6	1
1197.2.c.c		2	399.w	even	6	1
2128.2.m.a		2	28.f	even	6	1
2128.2.m.a		2	28.g	odd	6	1
2128.2.m.a		2	532.t	even	6	1
2128.2.m.a		2	532.bh	odd	6	1

Hecke kernels

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

\( T_{2} \)

\( T_{3} \)

\( T_{5}^{4} - 19T_{5}^{2} + 361 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( T^{4} \)
$5$	\( T^{4} - 19T^{2} + 361 \)
$7$	\( T^{4} \)
$11$	\( (T^{2} + 5 T + 25)^{2} \)