Newform orbit 931.2.o.d

• Label: 931.2.o.d
• Level: $931$
• Weight: $2$
• Character orbit: 931.o
• Analytic conductor: $7.434$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

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:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 133)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (z^3 - z^2 - z + 2) * q^2 + (2*z^3 + z^2 - z - 1) * q^3 + (4*z^3 - 2*z^2 - 2*z + 2) * q^4 + (2*z^3 - 2*z) * q^5 + 2*z^3 * q^6 + (6*z^3 - 4*z^2 + 2) * q^8 + (-2*z^3 - z^2 - 2*z) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(248\)	\(344\)
\(\chi(n)\)	\(\zeta_{12}^{2}\)	\(-1\)

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

0.866025	+	0.500000i
−0.866025	−	0.500000i
0.866025	−	0.500000i
−0.866025	+	0.500000i

0.633975

−

0.366025i

−1.36603

+

2.36603i

−0.732051

+

1.26795i

−1.73205

+

1.00000i

2.00000i

0

2.53590i

−2.23205

−

3.86603i

−0.732051

+

1.26795i

227.2

2.36603

−

1.36603i

0.366025

−

0.633975i

2.73205

−

4.73205i

1.73205

−

1.00000i

−

2.00000i

0

−

9.46410i

1.23205

+

2.13397i

2.73205

−

4.73205i

607.1

0.633975

+

0.366025i

−1.36603

−

2.36603i

−0.732051

−

1.26795i

−1.73205

−

1.00000i

−

2.00000i

0

−

2.53590i

−2.23205

+

3.86603i

−0.732051

−

1.26795i

607.2

2.36603

+

1.36603i

0.366025

+

0.633975i

2.73205

+

4.73205i

1.73205

+

1.00000i

2.00000i

0

9.46410i

1.23205

−

2.13397i

2.73205

+

4.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
133.o	even	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.d		4
7.b	odd	2	1		133.2.o.c	yes	4
7.c	even	3	1		133.2.o.a	✓	4
7.c	even	3	1		931.2.c.c		4
7.d	odd	6	1		931.2.c.a		4
7.d	odd	6	1		931.2.o.a		4
19.b	odd	2	1		931.2.o.a		4
133.c	even	2	1		133.2.o.a	✓	4
133.o	even	6	1		931.2.c.c		4
133.o	even	6	1	inner	931.2.o.d		4
133.r	odd	6	1		133.2.o.c	yes	4
133.r	odd	6	1		931.2.c.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
133.2.o.a	✓	4	7.c	even	3	1
133.2.o.a	✓	4	133.c	even	2	1
133.2.o.c	yes	4	7.b	odd	2	1
133.2.o.c	yes	4	133.r	odd	6	1
931.2.c.a		4	7.d	odd	6	1
931.2.c.a		4	133.r	odd	6	1
931.2.c.c		4	7.c	even	3	1
931.2.c.c		4	133.o	even	6	1
931.2.o.a		4	7.d	odd	6	1
931.2.o.a		4	19.b	odd	2	1
931.2.o.d		4	1.a	even	1	1	trivial
931.2.o.d		4	133.o	even	6	1	inner

Hecke kernels

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

\( T_{2}^{4} - 6T_{2}^{3} + 14T_{2}^{2} - 12T_{2} + 4 \)

\( T_{3}^{4} + 2T_{3}^{3} + 6T_{3}^{2} - 4T_{3} + 4 \)

\( T_{5}^{4} - 4T_{5}^{2} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 - 6*T^3 + 14*T^2 - 12*T + 4
$3$	T^4 + 2*T^3 + 6*T^2 - 4*T + 4
$5$	\( T^{4} - 4T^{2} + 16 \)
$7$	\( T^{4} \)
$11$	\( (T^{2} + 3 T + 9)^{2} \)