Newform orbit 507.2.f.d

• Label: 507.2.f.d
• Level: $507$
• Weight: $2$
• Character orbit: 507.f
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $8$
• CM discriminant: -39
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	8.0.56070144.2
Defining polynomial:	\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + \beta_{4} q^{3} + ( - \beta_{5} - 2 \beta_{2}) q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_1) q^{6} + (\beta_{7} + \beta_{3}) q^{8} + 3 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) :

\(\beta_{1}\)	\(=\)	\( ( 3\nu^{7} + 8\nu^{6} - 22\nu^{5} + 146\nu^{4} - 256\nu^{3} + 390\nu^{2} - 335\nu + 107 ) / 37 \)
\(\beta_{2}\)	\(=\)	\( ( 6\nu^{7} - 21\nu^{6} + 67\nu^{5} - 115\nu^{4} + 117\nu^{3} - 71\nu^{2} - 41\nu + 29 ) / 37 \)
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{7} + 29\nu^{6} - 89\nu^{5} + 261\nu^{4} - 373\nu^{3} + 461\nu^{2} - 220\nu + 41 ) / 37 \)
\(\beta_{4}\)	\(=\)	\( \nu^{6} - 3\nu^{5} + 11\nu^{4} - 17\nu^{3} + 25\nu^{2} - 17\nu + 9 \)
\(\beta_{5}\)	\(=\)	\( ( 16\nu^{7} - 56\nu^{6} + 228\nu^{5} - 430\nu^{4} + 756\nu^{3} - 732\nu^{2} + 532\nu - 157 ) / 37 \)
\(\beta_{6}\)	\(=\)	\( ( -42\nu^{7} + 147\nu^{6} - 543\nu^{5} + 953\nu^{4} - 1485\nu^{3} + 1163\nu^{2} - 749\nu + 56 ) / 37 \)
\(\beta_{7}\)	\(=\)	\( ( -42\nu^{7} + 147\nu^{6} - 543\nu^{5} + 1027\nu^{4} - 1633\nu^{3} + 1681\nu^{2} - 1193\nu + 500 ) / 37 \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(-1\)	\(\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

239.1

0.500000	−	2.19293i
0.500000	−	1.56488i
0.500000	+	0.564882i
0.500000	+	1.19293i
0.500000	+	2.19293i
0.500000	+	1.56488i
0.500000	−	0.564882i
0.500000	−	1.19293i

−1.69293

−

1.69293i

−1.73205

3.73205i

−1.23931

−

1.23931i

2.93225

+

2.93225i

0

2.93225

−

2.93225i

3.00000

4.19615i

239.2

−1.06488

−

1.06488i

1.73205

0.267949i

2.90931

+

2.90931i

−1.84443

−

1.84443i

0

−1.84443

+

1.84443i

3.00000

−

6.19615i

239.3

1.06488

+

1.06488i

1.73205

0.267949i

−2.90931

−

2.90931i

1.84443

+

1.84443i

0

1.84443

−

1.84443i

3.00000

−

6.19615i

239.4

1.69293

+

1.69293i

−1.73205

3.73205i

1.23931

+

1.23931i

−2.93225

−

2.93225i

0

−2.93225

+

2.93225i

3.00000

4.19615i

437.1

−1.69293

+

1.69293i

−1.73205

−

3.73205i

−1.23931

+

1.23931i

2.93225

−

2.93225i

0

2.93225

+

2.93225i

3.00000

−

4.19615i

437.2

−1.06488

+

1.06488i

1.73205

−

0.267949i

2.90931

−

2.90931i

−1.84443

+

1.84443i

0

−1.84443

−

1.84443i

3.00000

6.19615i

437.3

1.06488

−

1.06488i

1.73205

−

0.267949i

−2.90931

+

2.90931i

1.84443

−

1.84443i

0

1.84443

+

1.84443i

3.00000

6.19615i

437.4

1.69293

−

1.69293i

−1.73205

−

3.73205i

1.23931

−

1.23931i

−2.93225

+

2.93225i

0

−2.93225

−

2.93225i

3.00000

−

4.19615i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
39.d	odd	2	1	CM by \(\Q(\sqrt{-39}) \)
3.b	odd	2	1	inner
13.b	even	2	1	inner
13.d	odd	4	2	inner
39.f	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	507.2.f.d	✓	8
3.b	odd	2	1	inner	507.2.f.d	✓	8
13.b	even	2	1	inner	507.2.f.d	✓	8
13.c	even	3	1		507.2.k.g		8
13.c	even	3	1		507.2.k.h		8
13.d	odd	4	2	inner	507.2.f.d	✓	8
13.e	even	6	1		507.2.k.g		8
13.e	even	6	1		507.2.k.h		8
13.f	odd	12	2		507.2.k.g		8
13.f	odd	12	2		507.2.k.h		8
39.d	odd	2	1	CM	507.2.f.d	✓	8
39.f	even	4	2	inner	507.2.f.d	✓	8
39.h	odd	6	1		507.2.k.g		8
39.h	odd	6	1		507.2.k.h		8
39.i	odd	6	1		507.2.k.g		8
39.i	odd	6	1		507.2.k.h		8
39.k	even	12	2		507.2.k.g		8
39.k	even	12	2		507.2.k.h		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
507.2.f.d	✓	8	1.a	even	1	1	trivial
507.2.f.d	✓	8	3.b	odd	2	1	inner
507.2.f.d	✓	8	13.b	even	2	1	inner
507.2.f.d	✓	8	13.d	odd	4	2	inner
507.2.f.d	✓	8	39.d	odd	2	1	CM
507.2.f.d	✓	8	39.f	even	4	2	inner
507.2.k.g		8	13.c	even	3	1
507.2.k.g		8	13.e	even	6	1
507.2.k.g		8	13.f	odd	12	2
507.2.k.g		8	39.h	odd	6	1
507.2.k.g		8	39.i	odd	6	1
507.2.k.g		8	39.k	even	12	2
507.2.k.h		8	13.c	even	3	1
507.2.k.h		8	13.e	even	6	1
507.2.k.h		8	13.f	odd	12	2
507.2.k.h		8	39.h	odd	6	1
507.2.k.h		8	39.i	odd	6	1
507.2.k.h		8	39.k	even	12	2

Hecke kernels

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

\( T_{2}^{8} + 38T_{2}^{4} + 169 \)

\( T_{5}^{8} + 296T_{5}^{4} + 2704 \)

\( T_{7} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} + 38T^{4} + 169 \)
$3$	\( (T^{2} - 3)^{4} \)
$5$	\( T^{8} + 296T^{4} + 2704 \)
$7$	\( T^{8} \)
$11$	\( T^{8} + 1832 T^{4} + 2704 \)