Newform orbit 475.2.g.b

• Label: 475.2.g.b
• Level: $475$
• Weight: $2$
• Character orbit: 475.g
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 35x^{8} + 223x^{4} + 289 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{2} + \beta_{5} q^{3} + ( - \beta_{8} - \beta_{6}) q^{4} + (2 \beta_{4} + \beta_{2} - 2) q^{6} + ( - \beta_{8} + \beta_{6} + \beta_{2} + 1) q^{7} + \beta_{3} q^{8} + ( - \beta_{10} - \beta_{8} + 2 \beta_{6}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{6} + 12 q^{7}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 35x^{8} + 223x^{4} + 289 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{8} + 58\nu^{4} + 417 ) / 190 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{9} + 58\nu^{5} + 607\nu ) / 190 \)
\(\beta_{4}\)	\(=\)	\( ( -7\nu^{8} - 216\nu^{4} - 639 ) / 190 \)
\(\beta_{5}\)	\(=\)	\( ( -7\nu^{9} - 216\nu^{5} - 829\nu ) / 190 \)
\(\beta_{6}\)	\(=\)	\( ( 9\nu^{10} + 332\nu^{6} + 2993\nu^{2} ) / 3230 \)
\(\beta_{7}\)	\(=\)	\( ( 9\nu^{11} + 332\nu^{7} + 2993\nu^{3} ) / 3230 \)
\(\beta_{8}\)	\(=\)	\( ( -27\nu^{10} - 996\nu^{6} - 5749\nu^{2} ) / 3230 \)
\(\beta_{9}\)	\(=\)	\( ( -18\nu^{11} - 664\nu^{7} - 4371\nu^{3} ) / 1615 \)
\(\beta_{10}\)	\(=\)	\( ( -32\nu^{10} - 1001\nu^{6} - 3464\nu^{2} ) / 1615 \)
\(\beta_{11}\)	\(=\)	\( ( -73\nu^{11} - 2334\nu^{7} - 9921\nu^{3} ) / 3230 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(\beta_{6}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

18.1

−1.61467	+	1.61467i
−1.10924	+	1.10924i
−0.813901	+	0.813901i
0.813901	−	0.813901i
1.10924	−	1.10924i
1.61467	−	1.61467i
−1.61467	−	1.61467i
−1.10924	−	1.10924i
−0.813901	−	0.813901i
0.813901	+	0.813901i
1.10924	+	1.10924i
1.61467	+	1.61467i

−1.61467

−

1.61467i

1.11233

−

1.11233i

3.21432i

0

−3.59210

−1.21432

+

1.21432i

1.96073

−

1.96073i

0.525428i

0

18.2

−1.10924

−

1.10924i

−1.29790

+

1.29790i

0.460811i

0

2.87936

1.53919

−

1.53919i

−1.70732

+

1.70732i

−

0.369102i

0

18.3

−0.813901

−

0.813901i

2.01945

−

2.01945i

−

0.675131i

0

−3.28726

2.67513

−

2.67513i

−2.17729

+

2.17729i

−

5.15633i

0

18.4

0.813901

+

0.813901i

−2.01945

+

2.01945i

−

0.675131i

0

−3.28726

2.67513

−

2.67513i

2.17729

−

2.17729i

−

5.15633i

0

18.5

1.10924

+

1.10924i

1.29790

−

1.29790i

0.460811i

0

2.87936

1.53919

−

1.53919i

1.70732

−

1.70732i

−

0.369102i

0

18.6

1.61467

+

1.61467i

−1.11233

+

1.11233i

3.21432i

0

−3.59210

−1.21432

+

1.21432i

−1.96073

+

1.96073i

0.525428i

0

132.1

−1.61467

+

1.61467i

1.11233

+

1.11233i

−

3.21432i

0

−3.59210

−1.21432

−

1.21432i

1.96073

+

1.96073i

−

0.525428i

0

132.2

−1.10924

+

1.10924i

−1.29790

−

1.29790i

−

0.460811i

0

2.87936

1.53919

+

1.53919i

−1.70732

−

1.70732i

0.369102i

0

132.3

−0.813901

+

0.813901i

2.01945

+

2.01945i

0.675131i

0

−3.28726

2.67513

+

2.67513i

−2.17729

−

2.17729i

5.15633i

0

132.4

0.813901

−

0.813901i

−2.01945

−

2.01945i

0.675131i

0

−3.28726

2.67513

+

2.67513i

2.17729

+

2.17729i

5.15633i

0

132.5

1.10924

−

1.10924i

1.29790

+

1.29790i

−

0.460811i

0

2.87936

1.53919

+

1.53919i

1.70732

+

1.70732i

0.369102i

0

132.6

1.61467

−

1.61467i

−1.11233

−

1.11233i

−

3.21432i

0

−3.59210

−1.21432

−

1.21432i

−1.96073

−

1.96073i

−

0.525428i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
19.b	odd	2	1	inner
95.g	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	475.2.g.b		12
5.b	even	2	1		95.2.g.b	✓	12
5.c	odd	4	1		95.2.g.b	✓	12
5.c	odd	4	1	inner	475.2.g.b		12
15.d	odd	2	1		855.2.p.f		12
15.e	even	4	1		855.2.p.f		12
19.b	odd	2	1	inner	475.2.g.b		12
95.d	odd	2	1		95.2.g.b	✓	12
95.g	even	4	1		95.2.g.b	✓	12
95.g	even	4	1	inner	475.2.g.b		12
285.b	even	2	1		855.2.p.f		12
285.j	odd	4	1		855.2.p.f		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
95.2.g.b	✓	12	5.b	even	2	1
95.2.g.b	✓	12	5.c	odd	4	1
95.2.g.b	✓	12	95.d	odd	2	1
95.2.g.b	✓	12	95.g	even	4	1
475.2.g.b		12	1.a	even	1	1	trivial
475.2.g.b		12	5.c	odd	4	1	inner
475.2.g.b		12	19.b	odd	2	1	inner
475.2.g.b		12	95.g	even	4	1	inner
855.2.p.f		12	15.d	odd	2	1
855.2.p.f		12	15.e	even	4	1
855.2.p.f		12	285.b	even	2	1
855.2.p.f		12	285.j	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 35T_{2}^{8} + 223T_{2}^{4} + 289 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^12 + 35*T^8 + 223*T^4 + 289
$3$	T^12 + 84*T^8 + 1232*T^4 + 4624
$5$	\( T^{12} \)
$7$	(T^6 - 6*T^5 + 18*T^4 - 8*T^3 + 4*T^2 - 40*T + 200)^2
$11$	\( (T^{3} + 4 T^{2} - 4 T - 20)^{4} \)