Newform orbit 1950.2.y.c

• Label: 1950.2.y.c
• Level: $1950$
• Weight: $2$
• Character orbit: 1950.y
• Analytic conductor: $15.571$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.5708283941\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 390)
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.

\(f(q)\)	\(=\)	\( q - \zeta_{12}^{2} q^{2} + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{4} + \zeta_{12} q^{6} + ( - 2 \zeta_{12}^{2} + 2) q^{7} + q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\)	\(301\)	\(1301\)	\(1327\)
\(\chi(n)\)	\(1 - \zeta_{12}^{2}\)	\(1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

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

−0.500000

−

0.866025i

−0.866025

+

0.500000i

−0.500000

+

0.866025i

0

0.866025

+

0.500000i

1.00000

−

1.73205i

1.00000

0.500000

−

0.866025i

0

49.2

−0.500000

−

0.866025i

0.866025

−

0.500000i

−0.500000

+

0.866025i

0

−0.866025

−

0.500000i

1.00000

−

1.73205i

1.00000

0.500000

−

0.866025i

0

199.1

−0.500000

+

0.866025i

−0.866025

−

0.500000i

−0.500000

−

0.866025i

0

0.866025

−

0.500000i

1.00000

+

1.73205i

1.00000

0.500000

+

0.866025i

0

199.2

−0.500000

+

0.866025i

0.866025

+

0.500000i

−0.500000

−

0.866025i

0

−0.866025

+

0.500000i

1.00000

+

1.73205i

1.00000

0.500000

+

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1950.2.y.c		4
5.b	even	2	1		1950.2.y.f		4
5.c	odd	4	1		390.2.bb.b	✓	4
5.c	odd	4	1		1950.2.bc.b		4
13.e	even	6	1		1950.2.y.f		4
15.e	even	4	1		1170.2.bs.e		4
65.l	even	6	1	inner	1950.2.y.c		4
65.o	even	12	1		5070.2.a.bg		2
65.q	odd	12	1		5070.2.b.o		4
65.r	odd	12	1		390.2.bb.b	✓	4
65.r	odd	12	1		1950.2.bc.b		4
65.r	odd	12	1		5070.2.b.o		4
65.t	even	12	1		5070.2.a.y		2
195.bf	even	12	1		1170.2.bs.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.2.bb.b	✓	4	5.c	odd	4	1
390.2.bb.b	✓	4	65.r	odd	12	1
1170.2.bs.e		4	15.e	even	4	1
1170.2.bs.e		4	195.bf	even	12	1
1950.2.y.c		4	1.a	even	1	1	trivial
1950.2.y.c		4	65.l	even	6	1	inner
1950.2.y.f		4	5.b	even	2	1
1950.2.y.f		4	13.e	even	6	1
1950.2.bc.b		4	5.c	odd	4	1
1950.2.bc.b		4	65.r	odd	12	1
5070.2.a.y		2	65.t	even	12	1
5070.2.a.bg		2	65.o	even	12	1
5070.2.b.o		4	65.q	odd	12	1
5070.2.b.o		4	65.r	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 2T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{2} \)
$3$	\( T^{4} - T^{2} + 1 \)
$5$	\( T^{4} \)
$7$	\( (T^{2} - 2 T + 4)^{2} \)
$11$	\( T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9 \)