Newform orbit 950.2.j.g

• Label: 950.2.j.g
• Level: $950$
• Weight: $2$
• Character orbit: 950.j
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.49787136.1
Defining polynomial:	\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b4 * q^2 - b1 * q^3 + (b3 + 1) * q^4 + b7 * q^6 + (-b6 - b4 - b2 - b1) * q^7 + (b4 + b2) * q^8 + (4*b3 + 4) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} + 16 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{6} + 15\nu^{4} + 5\nu^{2} + 12 ) / 20 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} - 5\nu^{5} + 5\nu^{3} + 12\nu ) / 40 \)
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{6} - 5\nu^{4} - 15\nu^{2} - 36 ) / 20 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} - 7\nu ) / 10 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + 13\nu ) / 10 \)
\(\beta_{6}\)	\(=\)	\( ( -9\nu^{6} - 15\nu^{4} - 5\nu^{2} - 48 ) / 20 \)
\(\beta_{7}\)	\(=\)	\( ( 11\nu^{7} + 25\nu^{5} + 55\nu^{3} + 132\nu ) / 40 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(-1\)	\(-1 - \beta_{3}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

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

−0.866025

−

0.500000i

−2.29129

−

1.32288i

0.500000

+

0.866025i

0

1.32288

+

2.29129i

−

1.64575i

−

1.00000i

2.00000

+

3.46410i

0

49.2

−0.866025

−

0.500000i

2.29129

+

1.32288i

0.500000

+

0.866025i

0

−1.32288

−

2.29129i

3.64575i

−

1.00000i

2.00000

+

3.46410i

0

49.3

0.866025

+

0.500000i

−2.29129

−

1.32288i

0.500000

+

0.866025i

0

−1.32288

−

2.29129i

−

3.64575i

1.00000i

2.00000

+

3.46410i

0

49.4

0.866025

+

0.500000i

2.29129

+

1.32288i

0.500000

+

0.866025i

0

1.32288

+

2.29129i

1.64575i

1.00000i

2.00000

+

3.46410i

0

349.1

−0.866025

+

0.500000i

−2.29129

+

1.32288i

0.500000

−

0.866025i

0

1.32288

−

2.29129i

1.64575i

1.00000i

2.00000

−

3.46410i

0

349.2

−0.866025

+

0.500000i

2.29129

−

1.32288i

0.500000

−

0.866025i

0

−1.32288

+

2.29129i

−

3.64575i

1.00000i

2.00000

−

3.46410i

0

349.3

0.866025

−

0.500000i

−2.29129

+

1.32288i

0.500000

−

0.866025i

0

−1.32288

+

2.29129i

3.64575i

−

1.00000i

2.00000

−

3.46410i

0

349.4

0.866025

−

0.500000i

2.29129

−

1.32288i

0.500000

−

0.866025i

0

1.32288

−

2.29129i

−

1.64575i

−

1.00000i

2.00000

−

3.46410i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
19.c	even	3	1	inner
95.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	950.2.j.g		8
5.b	even	2	1	inner	950.2.j.g		8
5.c	odd	4	1		38.2.c.b	✓	4
5.c	odd	4	1		950.2.e.k		4
15.e	even	4	1		342.2.g.f		4
19.c	even	3	1	inner	950.2.j.g		8
20.e	even	4	1		304.2.i.e		4
40.i	odd	4	1		1216.2.i.l		4
40.k	even	4	1		1216.2.i.k		4
60.l	odd	4	1		2736.2.s.v		4
95.g	even	4	1		722.2.c.j		4
95.i	even	6	1	inner	950.2.j.g		8
95.l	even	12	1		722.2.a.g		2
95.l	even	12	1		722.2.c.j		4
95.m	odd	12	1		38.2.c.b	✓	4
95.m	odd	12	1		722.2.a.j		2
95.m	odd	12	1		950.2.e.k		4
95.q	odd	36	6		722.2.e.n		12
95.r	even	36	6		722.2.e.o		12
285.v	even	12	1		342.2.g.f		4
285.v	even	12	1		6498.2.a.ba		2
285.w	odd	12	1		6498.2.a.bg		2
380.v	even	12	1		304.2.i.e		4
380.v	even	12	1		5776.2.a.ba		2
380.w	odd	12	1		5776.2.a.z		2
760.br	odd	12	1		1216.2.i.l		4
760.bw	even	12	1		1216.2.i.k		4
1140.bu	odd	12	1		2736.2.s.v		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.2.c.b	✓	4	5.c	odd	4	1
38.2.c.b	✓	4	95.m	odd	12	1
304.2.i.e		4	20.e	even	4	1
304.2.i.e		4	380.v	even	12	1
342.2.g.f		4	15.e	even	4	1
342.2.g.f		4	285.v	even	12	1
722.2.a.g		2	95.l	even	12	1
722.2.a.j		2	95.m	odd	12	1
722.2.c.j		4	95.g	even	4	1
722.2.c.j		4	95.l	even	12	1
722.2.e.n		12	95.q	odd	36	6
722.2.e.o		12	95.r	even	36	6
950.2.e.k		4	5.c	odd	4	1
950.2.e.k		4	95.m	odd	12	1
950.2.j.g		8	1.a	even	1	1	trivial
950.2.j.g		8	5.b	even	2	1	inner
950.2.j.g		8	19.c	even	3	1	inner
950.2.j.g		8	95.i	even	6	1	inner
1216.2.i.k		4	40.k	even	4	1
1216.2.i.k		4	760.bw	even	12	1
1216.2.i.l		4	40.i	odd	4	1
1216.2.i.l		4	760.br	odd	12	1
2736.2.s.v		4	60.l	odd	4	1
2736.2.s.v		4	1140.bu	odd	12	1
5776.2.a.z		2	380.w	odd	12	1
5776.2.a.ba		2	380.v	even	12	1
6498.2.a.ba		2	285.v	even	12	1
6498.2.a.bg		2	285.w	odd	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{4} - 7T_{3}^{2} + 49 \)

\( T_{7}^{4} + 16T_{7}^{2} + 36 \)

\( T_{11}^{2} + 4T_{11} - 3 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{2} + 1)^{2} \)
$3$	\( (T^{4} - 7 T^{2} + 49)^{2} \)
$5$	\( T^{8} \)
$7$	\( (T^{4} + 16 T^{2} + 36)^{2} \)
$11$	\( (T^{2} + 4 T - 3)^{4} \)