Newform orbit 525.2.g.d

• Label: 525.2.g.d
• Level: $525$
• Weight: $2$
• Character orbit: 525.g
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.303595776.1
Defining polynomial:	\( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 105)
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.

\(f(q)\)	\(=\)	\( q + ( - \beta_{5} - \beta_{4} + \beta_1) q^{2} - \beta_1 q^{3} + (\beta_{7} - \beta_{2} + 1) q^{4} + ( - \beta_{7} - \beta_{2} - 1) q^{6} + (\beta_{6} + \beta_{4}) q^{7} + ( - \beta_{5} - 3 \beta_{4} + \beta_1) q^{8} + (\beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} - 4\nu^{4} + 16\nu^{2} + 9 ) / 36 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{6} + 4\nu^{4} + 20\nu^{2} + 27 ) / 36 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} - 4\nu^{5} - 2\nu^{3} + 9\nu ) / 54 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{7} + 5\nu^{5} + 16\nu^{3} + 45\nu ) / 27 \)
\(\beta_{6}\)	\(=\)	\( ( -5\nu^{7} - 16\nu^{5} - 8\nu^{3} - 81\nu ) / 108 \)
\(\beta_{7}\)	\(=\)	\( ( 5\nu^{6} + 16\nu^{4} + 44\nu^{2} + 117 ) / 36 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

524.1

−0.396143	+	1.68614i
−0.396143	−	1.68614i
−1.26217	+	1.18614i
−1.26217	−	1.18614i
1.26217	+	1.18614i
1.26217	−	1.18614i
0.396143	+	1.68614i
0.396143	−	1.68614i

−2.52434

0.396143

−

1.68614i

4.37228

0

−1.00000

+

4.25639i

1.73205

−

2.00000i

−5.98844

−2.68614

−

1.33591i

0

524.2

−2.52434

0.396143

+

1.68614i

4.37228

0

−1.00000

−

4.25639i

1.73205

+

2.00000i

−5.98844

−2.68614

+

1.33591i

0

524.3

−0.792287

1.26217

−

1.18614i

−1.37228

0

−1.00000

+

0.939764i

−1.73205

+

2.00000i

2.67181

0.186141

−

2.99422i

0

524.4

−0.792287

1.26217

+

1.18614i

−1.37228

0

−1.00000

−

0.939764i

−1.73205

−

2.00000i

2.67181

0.186141

+

2.99422i

0

524.5

0.792287

−1.26217

−

1.18614i

−1.37228

0

−1.00000

−

0.939764i

1.73205

+

2.00000i

−2.67181

0.186141

+

2.99422i

0

524.6

0.792287

−1.26217

+

1.18614i

−1.37228

0

−1.00000

+

0.939764i

1.73205

−

2.00000i

−2.67181

0.186141

−

2.99422i

0

524.7

2.52434

−0.396143

−

1.68614i

4.37228

0

−1.00000

−

4.25639i

−1.73205

−

2.00000i

5.98844

−2.68614

+

1.33591i

0

524.8

2.52434

−0.396143

+

1.68614i

4.37228

0

−1.00000

+

4.25639i

−1.73205

+

2.00000i

5.98844

−2.68614

−

1.33591i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
21.c	even	2	1	inner
105.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.2.g.d		8
3.b	odd	2	1		525.2.g.e		8
5.b	even	2	1	inner	525.2.g.d		8
5.c	odd	4	1		105.2.b.d	yes	4
5.c	odd	4	1		525.2.b.e		4
7.b	odd	2	1		525.2.g.e		8
15.d	odd	2	1		525.2.g.e		8
15.e	even	4	1		105.2.b.c	✓	4
15.e	even	4	1		525.2.b.g		4
20.e	even	4	1		1680.2.f.g		4
21.c	even	2	1	inner	525.2.g.d		8
35.c	odd	2	1		525.2.g.e		8
35.f	even	4	1		105.2.b.c	✓	4
35.f	even	4	1		525.2.b.g		4
35.k	even	12	1		735.2.s.h		4
35.k	even	12	1		735.2.s.i		4
35.l	odd	12	1		735.2.s.g		4
35.l	odd	12	1		735.2.s.j		4
60.l	odd	4	1		1680.2.f.h		4
105.g	even	2	1	inner	525.2.g.d		8
105.k	odd	4	1		105.2.b.d	yes	4
105.k	odd	4	1		525.2.b.e		4
105.w	odd	12	1		735.2.s.g		4
105.w	odd	12	1		735.2.s.j		4
105.x	even	12	1		735.2.s.h		4
105.x	even	12	1		735.2.s.i		4
140.j	odd	4	1		1680.2.f.h		4
420.w	even	4	1		1680.2.f.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.b.c	✓	4	15.e	even	4	1
105.2.b.c	✓	4	35.f	even	4	1
105.2.b.d	yes	4	5.c	odd	4	1
105.2.b.d	yes	4	105.k	odd	4	1
525.2.b.e		4	5.c	odd	4	1
525.2.b.e		4	105.k	odd	4	1
525.2.b.g		4	15.e	even	4	1
525.2.b.g		4	35.f	even	4	1
525.2.g.d		8	1.a	even	1	1	trivial
525.2.g.d		8	5.b	even	2	1	inner
525.2.g.d		8	21.c	even	2	1	inner
525.2.g.d		8	105.g	even	2	1	inner
525.2.g.e		8	3.b	odd	2	1
525.2.g.e		8	7.b	odd	2	1
525.2.g.e		8	15.d	odd	2	1
525.2.g.e		8	35.c	odd	2	1
735.2.s.g		4	35.l	odd	12	1
735.2.s.g		4	105.w	odd	12	1
735.2.s.h		4	35.k	even	12	1
735.2.s.h		4	105.x	even	12	1
735.2.s.i		4	35.k	even	12	1
735.2.s.i		4	105.x	even	12	1
735.2.s.j		4	35.l	odd	12	1
735.2.s.j		4	105.w	odd	12	1
1680.2.f.g		4	20.e	even	4	1
1680.2.f.g		4	420.w	even	4	1
1680.2.f.h		4	60.l	odd	4	1
1680.2.f.h		4	140.j	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} - 7T_{2}^{2} + 4 \)

\( T_{41} + 6 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - 7 T^{2} + 4)^{2} \)
$3$	T^8 + 5*T^6 + 16*T^4 + 45*T^2 + 81
$5$	\( T^{8} \)
$7$	\( (T^{4} + 2 T^{2} + 49)^{2} \)
$11$	\( (T^{4} + 7 T^{2} + 4)^{2} \)