Newform orbit 520.2.p.a

• Label: 520.2.p.a
• Level: $520$
• Weight: $2$
• Character orbit: 520.p
• Analytic conductor: $4.152$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.15222090511\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.207360000.1
Defining polynomial:	\( x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{7} - \beta_{4}) q^{3} - \beta_{5} q^{4} + ( - \beta_{6} - \beta_{2} - \beta_1) q^{5} + (2 \beta_{6} + \beta_{3}) q^{6} + (2 \beta_{2} + \beta_1) q^{8} + 7 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} + 56 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{6} + 8\nu^{4} + 48\nu^{2} + 56 ) / 32 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} + 4\nu^{4} + 24\nu^{2} - 8 ) / 16 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} + 8\nu^{5} + 48\nu^{3} + 120\nu ) / 32 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} + 4\nu^{5} + 24\nu^{3} - 24\nu ) / 16 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{6} - 6\nu^{4} - 28\nu^{2} - 16 ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( -3\nu^{7} - 16\nu^{5} - 80\nu^{3} - 8\nu ) / 32 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{7} - 6\nu^{5} - 32\nu^{3} - 16\nu ) / 8 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).

\(n\)	\(41\)	\(261\)	\(391\)	\(417\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

389.1

−1.14412	+	1.98168i
1.14412	−	1.98168i
−1.14412	−	1.98168i
1.14412	+	1.98168i
−0.437016	+	0.756934i
0.437016	−	0.756934i
−0.437016	−	0.756934i
0.437016	+	0.756934i

−1.11803

−

0.866025i

−3.16228

0.500000

+

1.93649i

−1.41421

+

1.73205i

3.53553

+

2.73861i

0

1.11803

−

2.59808i

7.00000

3.08114

−

0.711747i

389.2

−1.11803

−

0.866025i

3.16228

0.500000

+

1.93649i

1.41421

+

1.73205i

−3.53553

−

2.73861i

0

1.11803

−

2.59808i

7.00000

−0.0811388

−

3.16124i

389.3

−1.11803

+

0.866025i

−3.16228

0.500000

−

1.93649i

−1.41421

−

1.73205i

3.53553

−

2.73861i

0

1.11803

+

2.59808i

7.00000

3.08114

+

0.711747i

389.4

−1.11803

+

0.866025i

3.16228

0.500000

−

1.93649i

1.41421

−

1.73205i

−3.53553

+

2.73861i

0

1.11803

+

2.59808i

7.00000

−0.0811388

+

3.16124i

389.5

1.11803

−

0.866025i

−3.16228

0.500000

−

1.93649i

1.41421

+

1.73205i

−3.53553

+

2.73861i

0

−1.11803

−

2.59808i

7.00000

3.08114

+

0.711747i

389.6

1.11803

−

0.866025i

3.16228

0.500000

−

1.93649i

−1.41421

+

1.73205i

3.53553

−

2.73861i

0

−1.11803

−

2.59808i

7.00000

−0.0811388

+

3.16124i

389.7

1.11803

+

0.866025i

−3.16228

0.500000

+

1.93649i

1.41421

−

1.73205i

−3.53553

−

2.73861i

0

−1.11803

+

2.59808i

7.00000

3.08114

−

0.711747i

389.8

1.11803

+

0.866025i

3.16228

0.500000

+

1.93649i

−1.41421

−

1.73205i

3.53553

+

2.73861i

0

−1.11803

+

2.59808i

7.00000

−0.0811388

−

3.16124i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.b	even	2	1	inner
13.b	even	2	1	inner
40.f	even	2	1	inner
65.d	even	2	1	inner
104.e	even	2	1	inner
520.p	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	520.2.p.a	✓	8
4.b	odd	2	1		2080.2.p.a		8
5.b	even	2	1	inner	520.2.p.a	✓	8
8.b	even	2	1	inner	520.2.p.a	✓	8
8.d	odd	2	1		2080.2.p.a		8
13.b	even	2	1	inner	520.2.p.a	✓	8
20.d	odd	2	1		2080.2.p.a		8
40.e	odd	2	1		2080.2.p.a		8
40.f	even	2	1	inner	520.2.p.a	✓	8
52.b	odd	2	1		2080.2.p.a		8
65.d	even	2	1	inner	520.2.p.a	✓	8
104.e	even	2	1	inner	520.2.p.a	✓	8
104.h	odd	2	1		2080.2.p.a		8
260.g	odd	2	1		2080.2.p.a		8
520.b	odd	2	1		2080.2.p.a		8
520.p	even	2	1	inner	520.2.p.a	✓	8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
520.2.p.a	✓	8	1.a	even	1	1	trivial
520.2.p.a	✓	8	5.b	even	2	1	inner
520.2.p.a	✓	8	8.b	even	2	1	inner
520.2.p.a	✓	8	13.b	even	2	1	inner
520.2.p.a	✓	8	40.f	even	2	1	inner
520.2.p.a	✓	8	65.d	even	2	1	inner
520.2.p.a	✓	8	104.e	even	2	1	inner
520.2.p.a	✓	8	520.p	even	2	1	inner
2080.2.p.a		8	4.b	odd	2	1
2080.2.p.a		8	8.d	odd	2	1
2080.2.p.a		8	20.d	odd	2	1
2080.2.p.a		8	40.e	odd	2	1
2080.2.p.a		8	52.b	odd	2	1
2080.2.p.a		8	104.h	odd	2	1
2080.2.p.a		8	260.g	odd	2	1
2080.2.p.a		8	520.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 10 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{2} + 4)^{2} \)
$3$	\( (T^{2} - 10)^{4} \)
$5$	\( (T^{4} + 2 T^{2} + 25)^{2} \)
$7$	\( T^{8} \)
$11$	\( (T^{2} - 2)^{4} \)