Newform orbit 475.2.a.b

• Label: 475.2.a.b
• Level: $475$
• Weight: $2$
• Character orbit: 475.a
• Self dual: yes
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 2 q^{3} - 2 q^{4} + q^{7} + q^{9}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

0

0

2.00000

−2.00000

0

0

1.00000

0

1.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(1\)
\(19\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	475.2.a.b		1
3.b	odd	2	1		4275.2.a.i		1
4.b	odd	2	1		7600.2.a.c		1
5.b	even	2	1		19.2.a.a	✓	1
5.c	odd	4	2		475.2.b.a		2
15.d	odd	2	1		171.2.a.b		1
19.b	odd	2	1		9025.2.a.d		1
20.d	odd	2	1		304.2.a.f		1
35.c	odd	2	1		931.2.a.a		1
35.i	odd	6	2		931.2.f.b		2
35.j	even	6	2		931.2.f.c		2
40.e	odd	2	1		1216.2.a.b		1
40.f	even	2	1		1216.2.a.o		1
55.d	odd	2	1		2299.2.a.b		1
60.h	even	2	1		2736.2.a.c		1
65.d	even	2	1		3211.2.a.a		1
85.c	even	2	1		5491.2.a.b		1
95.d	odd	2	1		361.2.a.b		1
95.h	odd	6	2		361.2.c.a		2
95.i	even	6	2		361.2.c.c		2
95.o	odd	18	6		361.2.e.e		6
95.p	even	18	6		361.2.e.d		6
105.g	even	2	1		8379.2.a.j		1
285.b	even	2	1		3249.2.a.d		1
380.d	even	2	1		5776.2.a.c		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.2.a.a	✓	1	5.b	even	2	1
171.2.a.b		1	15.d	odd	2	1
304.2.a.f		1	20.d	odd	2	1
361.2.a.b		1	95.d	odd	2	1
361.2.c.a		2	95.h	odd	6	2
361.2.c.c		2	95.i	even	6	2
361.2.e.d		6	95.p	even	18	6
361.2.e.e		6	95.o	odd	18	6
475.2.a.b		1	1.a	even	1	1	trivial
475.2.b.a		2	5.c	odd	4	2
931.2.a.a		1	35.c	odd	2	1
931.2.f.b		2	35.i	odd	6	2
931.2.f.c		2	35.j	even	6	2
1216.2.a.b		1	40.e	odd	2	1
1216.2.a.o		1	40.f	even	2	1
2299.2.a.b		1	55.d	odd	2	1
2736.2.a.c		1	60.h	even	2	1
3211.2.a.a		1	65.d	even	2	1
3249.2.a.d		1	285.b	even	2	1
4275.2.a.i		1	3.b	odd	2	1
5491.2.a.b		1	85.c	even	2	1
5776.2.a.c		1	380.d	even	2	1
7600.2.a.c		1	4.b	odd	2	1
8379.2.a.j		1	105.g	even	2	1
9025.2.a.d		1	19.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\).

Hecke characteristic polynomials

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