Newform orbit 475.3.c.c

• Label: 475.3.c.c
• Level: $475$
• Weight: $3$
• Character orbit: 475.c
• Self dual: yes
• Analytic conductor: $12.943$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -19
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(12.9428125571\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{19}) \)
Defining polynomial:	\( x^{2} - 19 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 4 q^{4} - 3 \beta q^{7} + 9 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4} + 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

151.1

4.35890
−4.35890

0

0

4.00000

0

0

−13.0767

0

9.00000

0

151.2

0

0

4.00000

0

0

13.0767

0

9.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by \(\Q(\sqrt{-19}) \)
5.b	even	2	1	inner
95.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	475.3.c.c		2
5.b	even	2	1	inner	475.3.c.c		2
5.c	odd	4	2		95.3.d.a	✓	2
15.e	even	4	2		855.3.g.a		2
19.b	odd	2	1	CM	475.3.c.c		2
95.d	odd	2	1	inner	475.3.c.c		2
95.g	even	4	2		95.3.d.a	✓	2
285.j	odd	4	2		855.3.g.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
95.3.d.a	✓	2	5.c	odd	4	2
95.3.d.a	✓	2	95.g	even	4	2
475.3.c.c		2	1.a	even	1	1	trivial
475.3.c.c		2	5.b	even	2	1	inner
475.3.c.c		2	19.b	odd	2	1	CM
475.3.c.c		2	95.d	odd	2	1	inner
855.3.g.a		2	15.e	even	4	2
855.3.g.a		2	285.j	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\):

\( T_{2} \)

\( T_{7}^{2} - 171 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 171 \)
$11$	\( (T + 3)^{2} \)