Newform orbit 931.2.i.e

• Label: 931.2.i.e
• Level: $931$
• Weight: $2$
• Character orbit: 931.i
• 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.i (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:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial:	\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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 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.

\(f(q)\)	\(=\)	\( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{3} + \beta_1 - 1) q^{4} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{5} + (\beta_1 - 1) q^{6} + (2 \beta_{2} - 1) q^{8} + 2 \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 2 q^{4} - 3 q^{6} + 4 q^{9}+O(q^{10}) \)

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

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

411.1

−0.895644	−	1.09445i
1.39564	+	0.228425i
1.39564	−	0.228425i
−0.895644	+	1.09445i

−

2.18890i

0.500000

−

0.866025i

−2.79129

2.64575i

−1.89564

−

1.09445i

0

1.73205i

1.00000

+

1.73205i

5.79129

411.2

0.456850i

0.500000

−

0.866025i

1.79129

−

2.64575i

0.395644

+

0.228425i

0

1.73205i

1.00000

+

1.73205i

1.20871

521.1

−

0.456850i

0.500000

+

0.866025i

1.79129

2.64575i

0.395644

−

0.228425i

0

−

1.73205i

1.00000

−

1.73205i

1.20871

521.2

2.18890i

0.500000

+

0.866025i

−2.79129

−

2.64575i

−1.89564

+

1.09445i

0

−

1.73205i

1.00000

−

1.73205i

5.79129

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
133.i	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.i.e		4
7.b	odd	2	1		931.2.i.c		4
7.c	even	3	1		133.2.p.b	yes	4
7.c	even	3	1		931.2.s.d		4
7.d	odd	6	1		133.2.p.a	✓	4
7.d	odd	6	1		931.2.s.e		4
19.d	odd	6	1		931.2.s.e		4
133.i	even	6	1	inner	931.2.i.e		4
133.j	odd	6	1		931.2.i.c		4
133.n	odd	6	1		133.2.p.a	✓	4
133.p	even	6	1		931.2.s.d		4
133.s	even	6	1		133.2.p.b	yes	4

    

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

Hecke kernels

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

\( T_{2}^{4} + 5T_{2}^{2} + 1 \)

\( T_{3}^{2} - T_{3} + 1 \)

Hecke characteristic polynomials

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