LMFDB - Newform orbit 475.2.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

f(q)

=

q + 2 q^{3} - 2 q^{4} + q^{7} + q^{9} + 3 q^{11} - 4 q^{12} + 4 q^{13} + 4 q^{16} + 3 q^{17} + q^{19} + 2 q^{21} - 4 q^{27} - 2 q^{28} + 6 q^{29} - 4 q^{31} + 6 q^{33} - 2 q^{36} - 2 q^{37} + 8 q^{39}+ \cdots + 3 q^{99}+O(q^{100})

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

2.00000

−2.00000

1.00000

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 $T_{2}$ T2 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(475))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 2$ T - 2
$5$	$T$ T
$7$	$T - 1$ T - 1
$11$	$T - 3$ T - 3
$13$	$T - 4$ T - 4
$17$	$T - 3$ T - 3
$19$	$T - 1$ T - 1
$23$	$T$ T
$29$	$T - 6$ T - 6
$31$	$T + 4$ T + 4
$37$	$T + 2$ T + 2
$41$	$T + 6$ T + 6
$43$	$T - 1$ T - 1
$47$	$T - 3$ T - 3
$53$	$T + 12$ T + 12
$59$	$T + 6$ T + 6
$61$	$T + 1$ T + 1
$67$	$T - 4$ T - 4
$71$	$T - 6$ T - 6
$73$	$T - 7$ T - 7
$79$	$T - 8$ T - 8
$83$	$T + 12$ T + 12
$89$	$T - 12$ T - 12
$97$	$T + 8$ T + 8
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