LMFDB - Newform orbit 675.2.a.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, 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("675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$675 = 3^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	675.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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
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} + 3 q^{8} + 5 q^{11} - 5 q^{13} - q^{16} - 4 q^{17} - 2 q^{19} - 5 q^{22} + 3 q^{23} + 5 q^{26} + 10 q^{29} + 6 q^{31} - 5 q^{32} + 4 q^{34} + 5 q^{37} + 2 q^{38} + 10 q^{41} + 10 q^{43}+ \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

3.00000

Atkin-Lehner signs

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

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	675.2.a.c	yes	1
3.b	odd	2	1		675.2.a.g	yes	1
5.b	even	2	1		675.2.a.h	yes	1
5.c	odd	4	2		675.2.b.d		2
15.d	odd	2	1		675.2.a.b	✓	1
15.e	even	4	2		675.2.b.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
675.2.a.b	✓	1	15.d	odd	2	1
675.2.a.c	yes	1	1.a	even	1	1	trivial
675.2.a.g	yes	1	3.b	odd	2	1
675.2.a.h	yes	1	5.b	even	2	1
675.2.b.c		2	15.e	even	4	2
675.2.b.d		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(675))$ :

T_{2} + 1

T_{7}

T_{11} - 5

Hecke characteristic polynomials

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