Newform orbit 507.2.b.c

• Label: 507.2.b.c
• Level: $507$
• Weight: $2$
• Character orbit: 507.b
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\(x^{2} - x + 1\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{3} + 2 q^{4} + ( -2 + 4 \zeta_{6} ) q^{5} + ( 1 - 2 \zeta_{6} ) q^{7} + q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q - 2q^{3} + 4q^{4} + 2q^{9} + O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

337.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−1.00000

2.00000

−

3.46410i

0

1.73205i

0

1.00000

0

337.2

0

−1.00000

2.00000

3.46410i

0

−

1.73205i

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	507.2.b.c		2
3.b	odd	2	1		1521.2.b.f		2
13.b	even	2	1	inner	507.2.b.c		2
13.c	even	3	1		39.2.j.a	✓	2
13.c	even	3	1		507.2.j.b		2
13.d	odd	4	2		507.2.a.e		2
13.e	even	6	1		39.2.j.a	✓	2
13.e	even	6	1		507.2.j.b		2
13.f	odd	12	4		507.2.e.f		4
39.d	odd	2	1		1521.2.b.f		2
39.f	even	4	2		1521.2.a.h		2
39.h	odd	6	1		117.2.q.a		2
39.i	odd	6	1		117.2.q.a		2
52.f	even	4	2		8112.2.a.bu		2
52.i	odd	6	1		624.2.bv.b		2
52.j	odd	6	1		624.2.bv.b		2
65.l	even	6	1		975.2.bc.c		2
65.n	even	6	1		975.2.bc.c		2
65.q	odd	12	2		975.2.w.d		4
65.r	odd	12	2		975.2.w.d		4
156.p	even	6	1		1872.2.by.f		2
156.r	even	6	1		1872.2.by.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.j.a	✓	2	13.c	even	3	1
39.2.j.a	✓	2	13.e	even	6	1
117.2.q.a		2	39.h	odd	6	1
117.2.q.a		2	39.i	odd	6	1
507.2.a.e		2	13.d	odd	4	2
507.2.b.c		2	1.a	even	1	1	trivial
507.2.b.c		2	13.b	even	2	1	inner
507.2.e.f		4	13.f	odd	12	4
507.2.j.b		2	13.c	even	3	1
507.2.j.b		2	13.e	even	6	1
624.2.bv.b		2	52.i	odd	6	1
624.2.bv.b		2	52.j	odd	6	1
975.2.w.d		4	65.q	odd	12	2
975.2.w.d		4	65.r	odd	12	2
975.2.bc.c		2	65.l	even	6	1
975.2.bc.c		2	65.n	even	6	1
1521.2.a.h		2	39.f	even	4	2
1521.2.b.f		2	3.b	odd	2	1
1521.2.b.f		2	39.d	odd	2	1
1872.2.by.f		2	156.p	even	6	1
1872.2.by.f		2	156.r	even	6	1
8112.2.a.bu		2	52.f	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2} \)

\( T_{5}^{2} + 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( ( 1 + T )^{2} \)
$5$	\( 12 + T^{2} \)
$7$	\( 3 + T^{2} \)
$11$	\( 12 + T^{2} \)