Newform orbit 950.2.j.a

• Label: 950.2.j.a
• Level: $950$
• Weight: $2$
• Character orbit: 950.j
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $4$
• 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:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 190)
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.

\(f(q)\)	\(=\)	\( q - \zeta_{12} q^{2} + \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} - \zeta_{12}^{2} q^{6} - 2 \zeta_{12}^{3} q^{7} - \zeta_{12}^{3} q^{8} - 2 \zeta_{12}^{2} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{12}^{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

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

−0.866025

−

0.500000i

0.866025

+

0.500000i

0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−

2.00000i

−

1.00000i

−1.00000

−

1.73205i

0

49.2

0.866025

+

0.500000i

−0.866025

−

0.500000i

0.500000

+

0.866025i

0

−0.500000

−

0.866025i

2.00000i

1.00000i

−1.00000

−

1.73205i

0

349.1

−0.866025

+

0.500000i

0.866025

−

0.500000i

0.500000

−

0.866025i

0

−0.500000

+

0.866025i

2.00000i

1.00000i

−1.00000

+

1.73205i

0

349.2

0.866025

−

0.500000i

−0.866025

+

0.500000i

0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−

2.00000i

−

1.00000i

−1.00000

+

1.73205i

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.a		4
5.b	even	2	1	inner	950.2.j.a		4
5.c	odd	4	1		190.2.e.b	✓	2
5.c	odd	4	1		950.2.e.a		2
15.e	even	4	1		1710.2.l.b		2
19.c	even	3	1	inner	950.2.j.a		4
20.e	even	4	1		1520.2.q.e		2
95.i	even	6	1	inner	950.2.j.a		4
95.l	even	12	1		3610.2.a.i		1
95.m	odd	12	1		190.2.e.b	✓	2
95.m	odd	12	1		950.2.e.a		2
95.m	odd	12	1		3610.2.a.a		1
285.v	even	12	1		1710.2.l.b		2
380.v	even	12	1		1520.2.q.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
190.2.e.b	✓	2	5.c	odd	4	1
190.2.e.b	✓	2	95.m	odd	12	1
950.2.e.a		2	5.c	odd	4	1
950.2.e.a		2	95.m	odd	12	1
950.2.j.a		4	1.a	even	1	1	trivial
950.2.j.a		4	5.b	even	2	1	inner
950.2.j.a		4	19.c	even	3	1	inner
950.2.j.a		4	95.i	even	6	1	inner
1520.2.q.e		2	20.e	even	4	1
1520.2.q.e		2	380.v	even	12	1
1710.2.l.b		2	15.e	even	4	1
1710.2.l.b		2	285.v	even	12	1
3610.2.a.a		1	95.m	odd	12	1
3610.2.a.i		1	95.l	even	12	1

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{2} + 1 \)

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

\( T_{11} + 3 \)

Hecke characteristic polynomials

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