LMFDB - Newform orbit 2800.2.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2800 = 2^{4} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2800.g (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 350)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 3 i q^{3} - i q^{7} - 6 q^{9} + 5 q^{11} - 6 i q^{13} + i q^{17} - 3 q^{19} + 3 q^{21} - 9 i q^{27} + 6 q^{29} + 4 q^{31} + 15 i q^{33} - 8 i q^{37} + 18 q^{39} + 11 q^{41} + 8 i q^{43} + 2 i q^{47} + \cdots - 30 q^{99} +O(q^{100})$ q + 3i q^3 - i * q^7 - 6 * q^9 + 5 * q^11 - 6i q^13 + i * q^17 - 3 * q^19 + 3 * q^21 - 9i q^27 + 6 * q^29 + 4 * q^31 + 15i q^33 - 8i q^37 + 18 * q^39 + 11 * q^41 + 8i q^43 + 2i q^47 - q^49 - 3 * q^51 + 4i q^53 - 9i q^57 + 4 * q^59 - 2 * q^61 + 6i q^63 + 9i q^67 + 10 * q^71 - 7i q^73 - 5i q^77 - 2 * q^79 + 9 * q^81 - 11i q^83 + 18i q^87 + 11 * q^89 - 6 * q^91 + 12i q^93 + 10i q^97 - 30 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 12 q^{9} + 10 q^{11} - 6 q^{19} + 6 q^{21} + 12 q^{29} + 8 q^{31} + 36 q^{39} + 22 q^{41} - 2 q^{49} - 6 q^{51} + 8 q^{59} - 4 q^{61} + 20 q^{71} - 4 q^{79} + 18 q^{81} + 22 q^{89} - 12 q^{91} - 60 q^{99}+O(q^{100})$ 2 * q - 12 * q^9 + 10 * q^11 - 6 * q^19 + 6 * q^21 + 12 * q^29 + 8 * q^31 + 36 * q^39 + 22 * q^41 - 2 * q^49 - 6 * q^51 + 8 * q^59 - 4 * q^61 + 20 * q^71 - 4 * q^79 + 18 * q^81 + 22 * q^89 - 12 * q^91 - 60 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$ .

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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}

449.1

	−	1.00000i
		1.00000i

−

3.00000i

1.00000i

−6.00000

449.2

3.00000i

−

1.00000i

−6.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2800.2.g.a		2
4.b	odd	2	1		350.2.c.a		2
5.b	even	2	1	inner	2800.2.g.a		2
5.c	odd	4	1		2800.2.a.b		1
5.c	odd	4	1		2800.2.a.bg		1
12.b	even	2	1		3150.2.g.v		2
20.d	odd	2	1		350.2.c.a		2
20.e	even	4	1		350.2.a.c	✓	1
20.e	even	4	1		350.2.a.d	yes	1
28.d	even	2	1		2450.2.c.r		2
60.h	even	2	1		3150.2.g.v		2
60.l	odd	4	1		3150.2.a.j		1
60.l	odd	4	1		3150.2.a.bq		1
140.c	even	2	1		2450.2.c.r		2
140.j	odd	4	1		2450.2.a.a		1
140.j	odd	4	1		2450.2.a.bg		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
350.2.a.c	✓	1	20.e	even	4	1
350.2.a.d	yes	1	20.e	even	4	1
350.2.c.a		2	4.b	odd	2	1
350.2.c.a		2	20.d	odd	2	1
2450.2.a.a		1	140.j	odd	4	1
2450.2.a.bg		1	140.j	odd	4	1
2450.2.c.r		2	28.d	even	2	1
2450.2.c.r		2	140.c	even	2	1
2800.2.a.b		1	5.c	odd	4	1
2800.2.a.bg		1	5.c	odd	4	1
2800.2.g.a		2	1.a	even	1	1	trivial
2800.2.g.a		2	5.b	even	2	1	inner
3150.2.a.j		1	60.l	odd	4	1
3150.2.a.bq		1	60.l	odd	4	1
3150.2.g.v		2	12.b	even	2	1
3150.2.g.v		2	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 $S_{2}^{\mathrm{new}}(2800, [\chi])$ :

T_{3}^{2} + 9

T_{11} - 5

T_{13}^{2} + 36

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2}$ T^2
$7$	$T^{2} + 1$ T^2 + 1
$11$	$(T - 5)^{2}$ (T - 5)^2
$13$	$T^{2} + 36$ T^2 + 36
$17$	$T^{2} + 1$ T^2 + 1
$19$	$(T + 3)^{2}$ (T + 3)^2
$23$	$T^{2}$ T^2
$29$	$(T - 6)^{2}$ (T - 6)^2
$31$	$(T - 4)^{2}$ (T - 4)^2
$37$	$T^{2} + 64$ T^2 + 64
$41$	$(T - 11)^{2}$ (T - 11)^2
$43$	$T^{2} + 64$ T^2 + 64
$47$	$T^{2} + 4$ T^2 + 4
$53$	$T^{2} + 16$ T^2 + 16
$59$	$(T - 4)^{2}$ (T - 4)^2
$61$	$(T + 2)^{2}$ (T + 2)^2
$67$	$T^{2} + 81$ T^2 + 81
$71$	$(T - 10)^{2}$ (T - 10)^2
$73$	$T^{2} + 49$ T^2 + 49
$79$	$(T + 2)^{2}$ (T + 2)^2
$83$	$T^{2} + 121$ T^2 + 121
$89$	$(T - 11)^{2}$ (T - 11)^2
$97$	$T^{2} + 100$ T^2 + 100
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