LMFDB - Newform orbit 6300.2.a.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(1,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6300.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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 420)
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^{7} + 4 q^{11} + 6 q^{13} + 2 q^{17} + 6 q^{19} + 2 q^{23} - 6 q^{29} - 2 q^{31} + 4 q^{37} - 8 q^{41} + 4 q^{43} + 4 q^{47} + q^{49} + 6 q^{53} - 4 q^{59} + 14 q^{61} - 4 q^{67} + 10 q^{73} - 4 q^{77}+ \cdots - 10 q^{97}+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

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$5$	$-1$
$7$	$+1$

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	6300.2.a.n		1
3.b	odd	2	1		2100.2.a.j		1
5.b	even	2	1		6300.2.a.bc		1
5.c	odd	4	2		1260.2.k.d		2
12.b	even	2	1		8400.2.a.bh		1
15.d	odd	2	1		2100.2.a.e		1
15.e	even	4	2		420.2.k.a	✓	2
20.e	even	4	2		5040.2.t.o		2
60.h	even	2	1		8400.2.a.cd		1
60.l	odd	4	2		1680.2.t.a		2
105.k	odd	4	2		2940.2.k.d		2
105.w	odd	12	4		2940.2.bb.c		4
105.x	even	12	4		2940.2.bb.h		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
420.2.k.a	✓	2	15.e	even	4	2
1260.2.k.d		2	5.c	odd	4	2
1680.2.t.a		2	60.l	odd	4	2
2100.2.a.e		1	15.d	odd	2	1
2100.2.a.j		1	3.b	odd	2	1
2940.2.k.d		2	105.k	odd	4	2
2940.2.bb.c		4	105.w	odd	12	4
2940.2.bb.h		4	105.x	even	12	4
5040.2.t.o		2	20.e	even	4	2
6300.2.a.n		1	1.a	even	1	1	trivial
6300.2.a.bc		1	5.b	even	2	1
8400.2.a.bh		1	12.b	even	2	1
8400.2.a.cd		1	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6300))$ :

T_{11} - 4

T_{13} - 6

T_{17} - 2

T_{37} - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T$ T
$7$	$T + 1$ T + 1
$11$	$T - 4$ T - 4
$13$	$T - 6$ T - 6
$17$	$T - 2$ T - 2
$19$	$T - 6$ T - 6
$23$	$T - 2$ T - 2
$29$	$T + 6$ T + 6
$31$	$T + 2$ T + 2
$37$	$T - 4$ T - 4
$41$	$T + 8$ T + 8
$43$	$T - 4$ T - 4
$47$	$T - 4$ T - 4
$53$	$T - 6$ T - 6
$59$	$T + 4$ T + 4
$61$	$T - 14$ T - 14
$67$	$T + 4$ T + 4
$71$	$T$ T
$73$	$T - 10$ T - 10
$79$	$T$ T
$83$	$T + 16$ T + 16
$89$	$T + 8$ T + 8
$97$	$T + 10$ T + 10
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more