LMFDB - Newform orbit 1260.2.k.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1260,2,Mod(1009,1260)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1260, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1260.1009"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1260.k (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.0611506547$
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 140)
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.

$f(q)$	$=$	$q + ( - 2 i - 1) q^{5} + i q^{7} - 4 i q^{13} - 4 i q^{17} - 4 q^{19} + 8 i q^{23} + (4 i - 3) q^{25} + 2 q^{29} - 8 q^{31} + ( - i + 2) q^{35} - 8 i q^{37} - 6 q^{41} - 8 i q^{43} - 8 i q^{47} - q^{49} + \cdots - 12 i q^{97} +O(q^{100})$ q + (-2i - 1) q^5 + i * q^7 - 4i q^13 - 4i q^17 - 4 * q^19 + 8i q^23 + (4i - 3) q^25 + 2 * q^29 - 8 * q^31 + (-i + 2) * q^35 - 8i q^37 - 6 * q^41 - 8i q^43 - 8i q^47 - q^49 - 4 * q^59 - 6 * q^61 + (4i - 8) q^65 + 8i q^67 - 12 * q^71 + 4i q^73 + 4 * q^79 + (4i - 8) q^85 - 10 * q^89 + 4 * q^91 + (8i + 4) q^95 - 12i q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{5} - 8 q^{19} - 6 q^{25} + 4 q^{29} - 16 q^{31} + 4 q^{35} - 12 q^{41} - 2 q^{49} - 8 q^{59} - 12 q^{61} - 16 q^{65} - 24 q^{71} + 8 q^{79} - 16 q^{85} - 20 q^{89} + 8 q^{91} + 8 q^{95}+O(q^{100})$ 2 * q - 2 * q^5 - 8 * q^19 - 6 * q^25 + 4 * q^29 - 16 * q^31 + 4 * q^35 - 12 * q^41 - 2 * q^49 - 8 * q^59 - 12 * q^61 - 16 * q^65 - 24 * q^71 + 8 * q^79 - 16 * q^85 - 20 * q^89 + 8 * q^91 + 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$ .

$n$	$281$	$631$	$757$	$1081$
$\chi(n)$	$1$	$1$	$-1$	$1$

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}

1009.1

		1.00000i
	−	1.00000i

−1.00000

−

2.00000i

1.00000i

1009.2

−1.00000

2.00000i

−

1.00000i

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	1260.2.k.b		2
3.b	odd	2	1		140.2.e.b	✓	2
4.b	odd	2	1		5040.2.t.g		2
5.b	even	2	1	inner	1260.2.k.b		2
5.c	odd	4	1		6300.2.a.g		1
5.c	odd	4	1		6300.2.a.y		1
12.b	even	2	1		560.2.g.c		2
15.d	odd	2	1		140.2.e.b	✓	2
15.e	even	4	1		700.2.a.f		1
15.e	even	4	1		700.2.a.h		1
20.d	odd	2	1		5040.2.t.g		2
21.c	even	2	1		980.2.e.a		2
21.g	even	6	2		980.2.q.e		4
21.h	odd	6	2		980.2.q.d		4
24.f	even	2	1		2240.2.g.c		2
24.h	odd	2	1		2240.2.g.d		2
60.h	even	2	1		560.2.g.c		2
60.l	odd	4	1		2800.2.a.o		1
60.l	odd	4	1		2800.2.a.s		1
105.g	even	2	1		980.2.e.a		2
105.k	odd	4	1		4900.2.a.l		1
105.k	odd	4	1		4900.2.a.m		1
105.o	odd	6	2		980.2.q.d		4
105.p	even	6	2		980.2.q.e		4
120.i	odd	2	1		2240.2.g.d		2
120.m	even	2	1		2240.2.g.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.e.b	✓	2	3.b	odd	2	1
140.2.e.b	✓	2	15.d	odd	2	1
560.2.g.c		2	12.b	even	2	1
560.2.g.c		2	60.h	even	2	1
700.2.a.f		1	15.e	even	4	1
700.2.a.h		1	15.e	even	4	1
980.2.e.a		2	21.c	even	2	1
980.2.e.a		2	105.g	even	2	1
980.2.q.d		4	21.h	odd	6	2
980.2.q.d		4	105.o	odd	6	2
980.2.q.e		4	21.g	even	6	2
980.2.q.e		4	105.p	even	6	2
1260.2.k.b		2	1.a	even	1	1	trivial
1260.2.k.b		2	5.b	even	2	1	inner
2240.2.g.c		2	24.f	even	2	1
2240.2.g.c		2	120.m	even	2	1
2240.2.g.d		2	24.h	odd	2	1
2240.2.g.d		2	120.i	odd	2	1
2800.2.a.o		1	60.l	odd	4	1
2800.2.a.s		1	60.l	odd	4	1
4900.2.a.l		1	105.k	odd	4	1
4900.2.a.m		1	105.k	odd	4	1
5040.2.t.g		2	4.b	odd	2	1
5040.2.t.g		2	20.d	odd	2	1
6300.2.a.g		1	5.c	odd	4	1
6300.2.a.y		1	5.c	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{11}$ T11 acting on $S_{2}^{\mathrm{new}}(1260, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} + 2T + 5$ T^2 + 2*T + 5
$7$	$T^{2} + 1$ T^2 + 1
$11$	$T^{2}$ T^2
$13$	$T^{2} + 16$ T^2 + 16
$17$	$T^{2} + 16$ T^2 + 16
$19$	$(T + 4)^{2}$ (T + 4)^2
$23$	$T^{2} + 64$ T^2 + 64
$29$	$(T - 2)^{2}$ (T - 2)^2
$31$	$(T + 8)^{2}$ (T + 8)^2
$37$	$T^{2} + 64$ T^2 + 64
$41$	$(T + 6)^{2}$ (T + 6)^2
$43$	$T^{2} + 64$ T^2 + 64
$47$	$T^{2} + 64$ T^2 + 64
$53$	$T^{2}$ T^2
$59$	$(T + 4)^{2}$ (T + 4)^2
$61$	$(T + 6)^{2}$ (T + 6)^2
$67$	$T^{2} + 64$ T^2 + 64
$71$	$(T + 12)^{2}$ (T + 12)^2
$73$	$T^{2} + 16$ T^2 + 16
$79$	$(T - 4)^{2}$ (T - 4)^2
$83$	$T^{2}$ T^2
$89$	$(T + 10)^{2}$ (T + 10)^2
$97$	$T^{2} + 144$ T^2 + 144
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 $\F_{q}$
Belyi maps

$n$ :		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