LMFDB - Newform orbit 2400.4.a.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2400 = 2^{5} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 96)
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^{3} - 12 q^{7} + 9 q^{9} + 60 q^{11} + 42 q^{13} - 10 q^{17} + 132 q^{19} + 36 q^{21} + 48 q^{23} - 27 q^{27} + 226 q^{29} - 252 q^{31} - 180 q^{33} + 362 q^{37} - 126 q^{39} - 94 q^{41} + 228 q^{43}+ \cdots + 540 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

−3.00000

−12.0000

9.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$+1$
$5$	$+1$

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	2400.4.a.c		1
4.b	odd	2	1		2400.4.a.t		1
5.b	even	2	1		96.4.a.e	yes	1
15.d	odd	2	1		288.4.a.f		1
20.d	odd	2	1		96.4.a.b	✓	1
40.e	odd	2	1		192.4.a.j		1
40.f	even	2	1		192.4.a.d		1
60.h	even	2	1		288.4.a.e		1
80.k	odd	4	2		768.4.d.m		2
80.q	even	4	2		768.4.d.d		2
120.i	odd	2	1		576.4.a.o		1
120.m	even	2	1		576.4.a.n		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.4.a.b	✓	1	20.d	odd	2	1
96.4.a.e	yes	1	5.b	even	2	1
192.4.a.d		1	40.f	even	2	1
192.4.a.j		1	40.e	odd	2	1
288.4.a.e		1	60.h	even	2	1
288.4.a.f		1	15.d	odd	2	1
576.4.a.n		1	120.m	even	2	1
576.4.a.o		1	120.i	odd	2	1
768.4.d.d		2	80.q	even	4	2
768.4.d.m		2	80.k	odd	4	2
2400.4.a.c		1	1.a	even	1	1	trivial
2400.4.a.t		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2400))$ :

T_{7} + 12

T_{11} - 60

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 3$ T + 3
$5$	$T$ T
$7$	$T + 12$ T + 12
$11$	$T - 60$ T - 60
$13$	$T - 42$ T - 42
$17$	$T + 10$ T + 10
$19$	$T - 132$ T - 132
$23$	$T - 48$ T - 48
$29$	$T - 226$ T - 226
$31$	$T + 252$ T + 252
$37$	$T - 362$ T - 362
$41$	$T + 94$ T + 94
$43$	$T - 228$ T - 228
$47$	$T - 408$ T - 408
$53$	$T + 346$ T + 346
$59$	$T + 300$ T + 300
$61$	$T + 466$ T + 466
$67$	$T + 204$ T + 204
$71$	$T - 1056$ T - 1056
$73$	$T + 330$ T + 330
$79$	$T - 612$ T - 612
$83$	$T + 564$ T + 564
$89$	$T + 1510$ T + 1510
$97$	$T + 594$ T + 594
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