LMFDB - Newform orbit 980.1.f.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,1,Mod(99,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("980.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	980.f (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$0.489083712380$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.980.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.3841600.1

$q$-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^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8}+O(q^{10}) $

$ q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} - q^{10} + q^{12} + q^{15} + q^{16} + q^{20} + q^{23} - q^{24} + q^{25} - q^{27} - q^{29} - q^{30} - q^{32} - q^{40} - q^{41} + q^{43} - q^{46} - 2 q^{47} + q^{48} - q^{50} + q^{54} + q^{58} + q^{60} - q^{61} + q^{64} + q^{67} + q^{69} + q^{75} + q^{80} - q^{81} + q^{82} + q^{83} - q^{86} - q^{87} - q^{89} + q^{92} + 2 q^{94} - q^{96}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$1$	$-1$	$-1$

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} $

99.1

−1.00000

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	980.1.f.b		1
4.b	odd	2	1		980.1.f.c		1
5.b	even	2	1		980.1.f.c		1
7.b	odd	2	1		980.1.f.a		1
7.c	even	3	2		140.1.p.b	yes	2
7.d	odd	6	2		980.1.p.b		2
20.d	odd	2	1	CM	980.1.f.b		1
21.h	odd	6	2		1260.1.ci.a		2
28.d	even	2	1		980.1.f.d		1
28.f	even	6	2		980.1.p.a		2
28.g	odd	6	2		140.1.p.a	✓	2
35.c	odd	2	1		980.1.f.d		1
35.i	odd	6	2		980.1.p.a		2
35.j	even	6	2		140.1.p.a	✓	2
35.l	odd	12	4		700.1.u.a		4
56.k	odd	6	2		2240.1.bt.a		2
56.p	even	6	2		2240.1.bt.b		2
84.n	even	6	2		1260.1.ci.b		2
105.o	odd	6	2		1260.1.ci.b		2
140.c	even	2	1		980.1.f.a		1
140.p	odd	6	2		140.1.p.b	yes	2
140.s	even	6	2		980.1.p.b		2
140.w	even	12	4		700.1.u.a		4
280.bf	even	6	2		2240.1.bt.a		2
280.bi	odd	6	2		2240.1.bt.b		2
420.ba	even	6	2		1260.1.ci.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.1.p.a	✓	2	28.g	odd	6	2
140.1.p.a	✓	2	35.j	even	6	2
140.1.p.b	yes	2	7.c	even	3	2
140.1.p.b	yes	2	140.p	odd	6	2
700.1.u.a		4	35.l	odd	12	4
700.1.u.a		4	140.w	even	12	4
980.1.f.a		1	7.b	odd	2	1
980.1.f.a		1	140.c	even	2	1
980.1.f.b		1	1.a	even	1	1	trivial
980.1.f.b		1	20.d	odd	2	1	CM
980.1.f.c		1	4.b	odd	2	1
980.1.f.c		1	5.b	even	2	1
980.1.f.d		1	28.d	even	2	1
980.1.f.d		1	35.c	odd	2	1
980.1.p.a		2	28.f	even	6	2
980.1.p.a		2	35.i	odd	6	2
980.1.p.b		2	7.d	odd	6	2
980.1.p.b		2	140.s	even	6	2
1260.1.ci.a		2	21.h	odd	6	2
1260.1.ci.a		2	420.ba	even	6	2
1260.1.ci.b		2	84.n	even	6	2
1260.1.ci.b		2	105.o	odd	6	2
2240.1.bt.a		2	56.k	odd	6	2
2240.1.bt.a		2	280.bf	even	6	2
2240.1.bt.b		2	56.p	even	6	2
2240.1.bt.b		2	280.bi	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(980, [\chi])$:

$ T_{3} - 1 $

$ T_{23} - 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T - 1 $ T - 1
$5$	$ T - 1 $ T - 1
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T - 1 $ T - 1
$29$	$ T + 1 $ T + 1
$31$	$ T $ T
$37$	$ T $ T
$41$	$ T + 1 $ T + 1
$43$	$ T - 1 $ T - 1
$47$	$ T + 2 $ T + 2
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T + 1 $ T + 1
$67$	$ T - 1 $ T - 1
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T - 1 $ T - 1
$89$	$ T + 1 $ T + 1
$97$	$ T $ T
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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	980.f (of order \(2\), degree \(1\), minimal)

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more