Newform orbit 475.2.j.a

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

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( 2\zeta_{12} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{12}^{2} \)
\(\beta_{3}\)	\(=\)	\( 2\zeta_{12}^{3} \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(-1\)	\(-\beta_{2}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

0

−1.73205

−

1.00000i

−1.00000

−

1.73205i

0

0

−

4.00000i

0

0.500000

+

0.866025i

0

49.2

0

1.73205

+

1.00000i

−1.00000

−

1.73205i

0

0

4.00000i

0

0.500000

+

0.866025i

0

349.1

0

−1.73205

+

1.00000i

−1.00000

+

1.73205i

0

0

4.00000i

0

0.500000

−

0.866025i

0

349.2

0

1.73205

−

1.00000i

−1.00000

+

1.73205i

0

0

−

4.00000i

0

0.500000

−

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
19.c	even	3	1	inner
95.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	475.2.j.a		4
5.b	even	2	1	inner	475.2.j.a		4
5.c	odd	4	1		95.2.e.a	✓	2
5.c	odd	4	1		475.2.e.b		2
15.e	even	4	1		855.2.k.b		2
19.c	even	3	1	inner	475.2.j.a		4
20.e	even	4	1		1520.2.q.c		2
95.i	even	6	1	inner	475.2.j.a		4
95.l	even	12	1		1805.2.a.b		1
95.l	even	12	1		9025.2.a.e		1
95.m	odd	12	1		95.2.e.a	✓	2
95.m	odd	12	1		475.2.e.b		2
95.m	odd	12	1		1805.2.a.a		1
95.m	odd	12	1		9025.2.a.g		1
285.v	even	12	1		855.2.k.b		2
380.v	even	12	1		1520.2.q.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
95.2.e.a	✓	2	5.c	odd	4	1
95.2.e.a	✓	2	95.m	odd	12	1
475.2.e.b		2	5.c	odd	4	1
475.2.e.b		2	95.m	odd	12	1
475.2.j.a		4	1.a	even	1	1	trivial
475.2.j.a		4	5.b	even	2	1	inner
475.2.j.a		4	19.c	even	3	1	inner
475.2.j.a		4	95.i	even	6	1	inner
855.2.k.b		2	15.e	even	4	1
855.2.k.b		2	285.v	even	12	1
1520.2.q.c		2	20.e	even	4	1
1520.2.q.c		2	380.v	even	12	1
1805.2.a.a		1	95.m	odd	12	1
1805.2.a.b		1	95.l	even	12	1
9025.2.a.e		1	95.l	even	12	1
9025.2.a.g		1	95.m	odd	12	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( T^{4} - 4T^{2} + 16 \)
$5$	\( T^{4} \)
$7$	\( (T^{2} + 16)^{2} \)
$11$	\( (T - 3)^{4} \)