LMFDB - Newform orbit 585.2.a.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("585.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$585 = 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	585.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:	$4.67124851824$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 195)
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 + q^{2} - q^{4} - q^{5} - 3 q^{8} - q^{10} - 4 q^{11} + q^{13} - q^{16} - 2 q^{17} - 4 q^{19} + q^{20} - 4 q^{22} - 8 q^{23} + q^{25} + q^{26} + 2 q^{29} - 8 q^{31} + 5 q^{32} - 2 q^{34} + 6 q^{37}+ \cdots - 7 q^{98}+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

1.00000

−1.00000

−3.00000

−1.00000

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$+1$
$13$	$-1$

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	585.2.a.g		1
3.b	odd	2	1		195.2.a.a	✓	1
4.b	odd	2	1		9360.2.a.o		1
5.b	even	2	1		2925.2.a.d		1
5.c	odd	4	2		2925.2.c.f		2
12.b	even	2	1		3120.2.a.k		1
13.b	even	2	1		7605.2.a.h		1
15.d	odd	2	1		975.2.a.i		1
15.e	even	4	2		975.2.c.e		2
21.c	even	2	1		9555.2.a.b		1
39.d	odd	2	1		2535.2.a.k		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
195.2.a.a	✓	1	3.b	odd	2	1
585.2.a.g		1	1.a	even	1	1	trivial
975.2.a.i		1	15.d	odd	2	1
975.2.c.e		2	15.e	even	4	2
2535.2.a.k		1	39.d	odd	2	1
2925.2.a.d		1	5.b	even	2	1
2925.2.c.f		2	5.c	odd	4	2
3120.2.a.k		1	12.b	even	2	1
7605.2.a.h		1	13.b	even	2	1
9360.2.a.o		1	4.b	odd	2	1
9555.2.a.b		1	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(585))$ :

T_{2} - 1

T_{7}

T_{11} + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 1$ T - 1
$3$	$T$ T
$5$	$T + 1$ T + 1
$7$	$T$ T
$11$	$T + 4$ T + 4
$13$	$T - 1$ T - 1
$17$	$T + 2$ T + 2
$19$	$T + 4$ T + 4
$23$	$T + 8$ T + 8
$29$	$T - 2$ T - 2
$31$	$T + 8$ T + 8
$37$	$T - 6$ T - 6
$41$	$T - 6$ T - 6
$43$	$T + 4$ T + 4
$47$	$T - 8$ T - 8
$53$	$T + 6$ T + 6
$59$	$T - 12$ T - 12
$61$	$T + 2$ T + 2
$67$	$T + 4$ T + 4
$71$	$T$ T
$73$	$T + 6$ T + 6
$79$	$T - 16$ T - 16
$83$	$T - 4$ T - 4
$89$	$T + 10$ T + 10
$97$	$T - 18$ T - 18
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