LMFDB - Newform orbit 600.4.a.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(1,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, 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("600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$600 = 2^{3} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	600.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:	$35.4011460034$
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 + 3 q^{3} + 5 q^{7} + 9 q^{9} + 14 q^{11} + q^{13} + 46 q^{17} + 19 q^{19} + 15 q^{21} - 46 q^{23} + 27 q^{27} + 14 q^{29} + 133 q^{31} + 42 q^{33} + 258 q^{37} + 3 q^{39} + 84 q^{41} - 167 q^{43} + 410 q^{47}+ \cdots + 126 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

5.00000

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	600.4.a.n	yes	1
3.b	odd	2	1		1800.4.a.v		1
4.b	odd	2	1		1200.4.a.g		1
5.b	even	2	1		600.4.a.e	✓	1
5.c	odd	4	2		600.4.f.e		2
15.d	odd	2	1		1800.4.a.m		1
15.e	even	4	2		1800.4.f.l		2
20.d	odd	2	1		1200.4.a.bd		1
20.e	even	4	2		1200.4.f.i		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
600.4.a.e	✓	1	5.b	even	2	1
600.4.a.n	yes	1	1.a	even	1	1	trivial
600.4.f.e		2	5.c	odd	4	2
1200.4.a.g		1	4.b	odd	2	1
1200.4.a.bd		1	20.d	odd	2	1
1200.4.f.i		2	20.e	even	4	2
1800.4.a.m		1	15.d	odd	2	1
1800.4.a.v		1	3.b	odd	2	1
1800.4.f.l		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(600))$ :

T_{7} - 5

T_{11} - 14

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 3$ T - 3
$5$	$T$ T
$7$	$T - 5$ T - 5
$11$	$T - 14$ T - 14
$13$	$T - 1$ T - 1
$17$	$T - 46$ T - 46
$19$	$T - 19$ T - 19
$23$	$T + 46$ T + 46
$29$	$T - 14$ T - 14
$31$	$T - 133$ T - 133
$37$	$T - 258$ T - 258
$41$	$T - 84$ T - 84
$43$	$T + 167$ T + 167
$47$	$T - 410$ T - 410
$53$	$T - 456$ T - 456
$59$	$T + 194$ T + 194
$61$	$T + 17$ T + 17
$67$	$T - 653$ T - 653
$71$	$T - 828$ T - 828
$73$	$T - 570$ T - 570
$79$	$T + 552$ T + 552
$83$	$T - 142$ T - 142
$89$	$T + 1104$ T + 1104
$97$	$T - 841$ T - 841
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