LMFDB - Newform orbit 1575.4.a.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1575 = 3^{2} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 315)
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 + 3 q^{2} + q^{4} - 7 q^{7} - 21 q^{8} - 60 q^{11} - 38 q^{13} - 21 q^{14} - 71 q^{16} - 84 q^{17} + 110 q^{19} - 180 q^{22} + 120 q^{23} - 114 q^{26} - 7 q^{28} - 162 q^{29} + 236 q^{31} - 45 q^{32}+ \cdots + 147 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

3.00000

1.00000

−7.00000

−21.0000

Atkin-Lehner signs

$p$	Sign
$3$	$+1$
$5$	$+1$
$7$	$+1$

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	1575.4.a.i		1
3.b	odd	2	1		1575.4.a.c		1
5.b	even	2	1		315.4.a.b	✓	1
15.d	odd	2	1		315.4.a.e	yes	1
35.c	odd	2	1		2205.4.a.f		1
105.g	even	2	1		2205.4.a.p		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.4.a.b	✓	1	5.b	even	2	1
315.4.a.e	yes	1	15.d	odd	2	1
1575.4.a.c		1	3.b	odd	2	1
1575.4.a.i		1	1.a	even	1	1	trivial
2205.4.a.f		1	35.c	odd	2	1
2205.4.a.p		1	105.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1575))$ :

T_{2} - 3

T_{11} + 60

T_{13} + 38

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 3$ T - 3
$3$	$T$ T
$5$	$T$ T
$7$	$T + 7$ T + 7
$11$	$T + 60$ T + 60
$13$	$T + 38$ T + 38
$17$	$T + 84$ T + 84
$19$	$T - 110$ T - 110
$23$	$T - 120$ T - 120
$29$	$T + 162$ T + 162
$31$	$T - 236$ T - 236
$37$	$T - 376$ T - 376
$41$	$T - 126$ T - 126
$43$	$T - 34$ T - 34
$47$	$T + 6$ T + 6
$53$	$T - 582$ T - 582
$59$	$T + 492$ T + 492
$61$	$T + 880$ T + 880
$67$	$T - 826$ T - 826
$71$	$T - 666$ T - 666
$73$	$T - 826$ T - 826
$79$	$T + 592$ T + 592
$83$	$T - 792$ T - 792
$89$	$T + 1002$ T + 1002
$97$	$T + 1442$ T + 1442
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