Newform orbit 507.1.c.a

• Label: 507.1.c.a
• Level: $507$
• Weight: $1$
• Character orbit: 507.c
• Self dual: yes
• Analytic conductor: $0.253$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -3, -39, 13
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	507.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.253025961405\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.2.6591.1

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{3} + q^{4} + q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

170.1

0

0

−1.00000

1.00000

0

0

0

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
13.b	even	2	1	RM by \(\Q(\sqrt{13}) \)
39.d	odd	2	1	CM by \(\Q(\sqrt{-39}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	507.1.c.a		1
3.b	odd	2	1	CM	507.1.c.a		1
13.b	even	2	1	RM	507.1.c.a		1
13.c	even	3	2		507.1.i.a		2
13.d	odd	4	2		39.1.d.a	✓	1
13.e	even	6	2		507.1.i.a		2
13.f	odd	12	4		507.1.h.a		2
39.d	odd	2	1	CM	507.1.c.a		1
39.f	even	4	2		39.1.d.a	✓	1
39.h	odd	6	2		507.1.i.a		2
39.i	odd	6	2		507.1.i.a		2
39.k	even	12	4		507.1.h.a		2
52.f	even	4	2		624.1.l.a		1
65.f	even	4	2		975.1.e.a		2
65.g	odd	4	2		975.1.g.a		1
65.k	even	4	2		975.1.e.a		2
91.i	even	4	2		1911.1.h.a		1
91.z	odd	12	4		1911.1.w.b		2
91.bb	even	12	4		1911.1.w.a		2
104.j	odd	4	2		2496.1.l.b		1
104.m	even	4	2		2496.1.l.a		1
117.y	odd	12	4		1053.1.n.b		2
117.z	even	12	4		1053.1.n.b		2
156.l	odd	4	2		624.1.l.a		1
195.j	odd	4	2		975.1.e.a		2
195.n	even	4	2		975.1.g.a		1
195.u	odd	4	2		975.1.e.a		2
273.o	odd	4	2		1911.1.h.a		1
273.cb	odd	12	4		1911.1.w.a		2
273.cd	even	12	4		1911.1.w.b		2
312.w	odd	4	2		2496.1.l.a		1
312.y	even	4	2		2496.1.l.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.1.d.a	✓	1	13.d	odd	4	2
39.1.d.a	✓	1	39.f	even	4	2
507.1.c.a		1	1.a	even	1	1	trivial
507.1.c.a		1	3.b	odd	2	1	CM
507.1.c.a		1	13.b	even	2	1	RM
507.1.c.a		1	39.d	odd	2	1	CM
507.1.h.a		2	13.f	odd	12	4
507.1.h.a		2	39.k	even	12	4
507.1.i.a		2	13.c	even	3	2
507.1.i.a		2	13.e	even	6	2
507.1.i.a		2	39.h	odd	6	2
507.1.i.a		2	39.i	odd	6	2
624.1.l.a		1	52.f	even	4	2
624.1.l.a		1	156.l	odd	4	2
975.1.e.a		2	65.f	even	4	2
975.1.e.a		2	65.k	even	4	2
975.1.e.a		2	195.j	odd	4	2
975.1.e.a		2	195.u	odd	4	2
975.1.g.a		1	65.g	odd	4	2
975.1.g.a		1	195.n	even	4	2
1053.1.n.b		2	117.y	odd	12	4
1053.1.n.b		2	117.z	even	12	4
1911.1.h.a		1	91.i	even	4	2
1911.1.h.a		1	273.o	odd	4	2
1911.1.w.a		2	91.bb	even	12	4
1911.1.w.a		2	273.cb	odd	12	4
1911.1.w.b		2	91.z	odd	12	4
1911.1.w.b		2	273.cd	even	12	4
2496.1.l.a		1	104.m	even	4	2
2496.1.l.a		1	312.w	odd	4	2
2496.1.l.b		1	104.j	odd	4	2
2496.1.l.b		1	312.y	even	4	2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(507, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T + 1 \)
$5$	\( T \)
$7$	\( T \)
$11$	\( T \)