LMFDB - Newform orbit 2548.2.k.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,2,Mod(393,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("2548.393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2548 = 2^{2} \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2548.k (of order $3$ , degree $2$ , 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:	$20.3458824350$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 3 q^{5} + 3 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + (3 \zeta_{6} + 1) q^{13} - 7 \zeta_{6} q^{17} + 2 \zeta_{6} q^{19} + ( - 4 \zeta_{6} + 4) q^{23} + 4 q^{25} + (\zeta_{6} - 1) q^{29} - 4 q^{31} + \cdots + 6 q^{99} +O(q^{100})$ q - 3 * q^5 + 3z q^9 + (-2z + 2) q^11 + (3z + 1) q^13 - 7z q^17 + 2z q^19 + (-4z + 4) q^23 + 4 * q^25 + (z - 1) * q^29 - 4 * q^31 + (z - 1) * q^37 + (3z - 3) q^41 + 6z q^43 - 9z q^45 + 10 * q^47 - 7 * q^53 + (6z - 6) q^55 + 6z q^59 + 7z q^61 + (-9z - 3) q^65 + (8z - 8) q^67 + 6z q^71 + 11 * q^73 - 14 * q^79 + (9z - 9) q^81 + 14 * q^83 + 21z q^85 + (10z - 10) q^89 - 6z q^95 + 2z q^97 + 6 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 6 q^{5} + 3 q^{9} + 2 q^{11} + 5 q^{13} - 7 q^{17} + 2 q^{19} + 4 q^{23} + 8 q^{25} - q^{29} - 8 q^{31} - q^{37} - 3 q^{41} + 6 q^{43} - 9 q^{45} + 20 q^{47} - 14 q^{53} - 6 q^{55} + 6 q^{59} + 7 q^{61}+ \cdots + 12 q^{99}+O(q^{100})$ 2 * q - 6 * q^5 + 3 * q^9 + 2 * q^11 + 5 * q^13 - 7 * q^17 + 2 * q^19 + 4 * q^23 + 8 * q^25 - q^29 - 8 * q^31 - q^37 - 3 * q^41 + 6 * q^43 - 9 * q^45 + 20 * q^47 - 14 * q^53 - 6 * q^55 + 6 * q^59 + 7 * q^61 - 15 * q^65 - 8 * q^67 + 6 * q^71 + 22 * q^73 - 28 * q^79 - 9 * q^81 + 28 * q^83 + 21 * q^85 - 10 * q^89 - 6 * q^95 + 2 * q^97 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$ .

$n$	$197$	$885$	$1275$
$\chi(n)$	$-\zeta_{6}$	$1$	$1$

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}

393.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−3.00000

1.50000

−

2.59808i

1569.1

−3.00000

1.50000

2.59808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2548.2.k.b		2
7.b	odd	2	1		364.2.k.b	✓	2
7.c	even	3	1		2548.2.i.f		2
7.c	even	3	1		2548.2.l.f		2
7.d	odd	6	1		2548.2.i.c		2
7.d	odd	6	1		2548.2.l.c		2
13.c	even	3	1	inner	2548.2.k.b		2
21.c	even	2	1		3276.2.z.a		2
28.d	even	2	1		1456.2.s.d		2
91.g	even	3	1		2548.2.i.f		2
91.h	even	3	1		2548.2.l.f		2
91.m	odd	6	1		2548.2.i.c		2
91.n	odd	6	1		364.2.k.b	✓	2
91.n	odd	6	1		4732.2.a.f		1
91.t	odd	6	1		4732.2.a.b		1
91.v	odd	6	1		2548.2.l.c		2
91.bc	even	12	2		4732.2.g.d		2
273.bn	even	6	1		3276.2.z.a		2
364.v	even	6	1		1456.2.s.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
364.2.k.b	✓	2	7.b	odd	2	1
364.2.k.b	✓	2	91.n	odd	6	1
1456.2.s.d		2	28.d	even	2	1
1456.2.s.d		2	364.v	even	6	1
2548.2.i.c		2	7.d	odd	6	1
2548.2.i.c		2	91.m	odd	6	1
2548.2.i.f		2	7.c	even	3	1
2548.2.i.f		2	91.g	even	3	1
2548.2.k.b		2	1.a	even	1	1	trivial
2548.2.k.b		2	13.c	even	3	1	inner
2548.2.l.c		2	7.d	odd	6	1
2548.2.l.c		2	91.v	odd	6	1
2548.2.l.f		2	7.c	even	3	1
2548.2.l.f		2	91.h	even	3	1
3276.2.z.a		2	21.c	even	2	1
3276.2.z.a		2	273.bn	even	6	1
4732.2.a.b		1	91.t	odd	6	1
4732.2.a.f		1	91.n	odd	6	1
4732.2.g.d		2	91.bc	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2548, [\chi])$ :

T_{3}

T_{5} + 3

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$(T + 3)^{2}$ (T + 3)^2
$7$	$T^{2}$ T^2
$11$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$13$	$T^{2} - 5T + 13$ T^2 - 5*T + 13
$17$	$T^{2} + 7T + 49$ T^2 + 7*T + 49
$19$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$23$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$29$	$T^{2} + T + 1$ T^2 + T + 1
$31$	$(T + 4)^{2}$ (T + 4)^2
$37$	$T^{2} + T + 1$ T^2 + T + 1
$41$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$43$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$47$	$(T - 10)^{2}$ (T - 10)^2
$53$	$(T + 7)^{2}$ (T + 7)^2
$59$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$61$	$T^{2} - 7T + 49$ T^2 - 7*T + 49
$67$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$71$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$73$	$(T - 11)^{2}$ (T - 11)^2
$79$	$(T + 14)^{2}$ (T + 14)^2
$83$	$(T - 14)^{2}$ (T - 14)^2
$89$	$T^{2} + 10T + 100$ T^2 + 10*T + 100
$97$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
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