LMFDB - Newform orbit 198.4.a.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [198,4,Mod(1,198)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(198, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("198.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$198 = 2 \cdot 3^{2} \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	198.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:	$11.6823781811$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 66)
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 + 2 q^{2} + 4 q^{4} + 14 q^{7} + 8 q^{8} - 11 q^{11} + 80 q^{13} + 28 q^{14} + 16 q^{16} - 30 q^{17} + 56 q^{19} - 22 q^{22} + 126 q^{23} - 125 q^{25} + 160 q^{26} + 56 q^{28} + 222 q^{29} - 16 q^{31}+ \cdots - 294 q^{98}+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

2.00000

4.00000

14.0000

8.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$11$	$+1$

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	198.4.a.f		1
3.b	odd	2	1		66.4.a.a	✓	1
4.b	odd	2	1		1584.4.a.i		1
11.b	odd	2	1		2178.4.a.g		1
12.b	even	2	1		528.4.a.d		1
15.d	odd	2	1		1650.4.a.h		1
15.e	even	4	2		1650.4.c.g		2
24.f	even	2	1		2112.4.a.s		1
24.h	odd	2	1		2112.4.a.g		1
33.d	even	2	1		726.4.a.h		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
66.4.a.a	✓	1	3.b	odd	2	1
198.4.a.f		1	1.a	even	1	1	trivial
528.4.a.d		1	12.b	even	2	1
726.4.a.h		1	33.d	even	2	1
1584.4.a.i		1	4.b	odd	2	1
1650.4.a.h		1	15.d	odd	2	1
1650.4.c.g		2	15.e	even	4	2
2112.4.a.g		1	24.h	odd	2	1
2112.4.a.s		1	24.f	even	2	1
2178.4.a.g		1	11.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}$ T5 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(198))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 2$ T - 2
$3$	$T$ T
$5$	$T$ T
$7$	$T - 14$ T - 14
$11$	$T + 11$ T + 11
$13$	$T - 80$ T - 80
$17$	$T + 30$ T + 30
$19$	$T - 56$ T - 56
$23$	$T - 126$ T - 126
$29$	$T - 222$ T - 222
$31$	$T + 16$ T + 16
$37$	$T + 106$ T + 106
$41$	$T + 114$ T + 114
$43$	$T + 52$ T + 52
$47$	$T + 246$ T + 246
$53$	$T - 264$ T - 264
$59$	$T + 264$ T + 264
$61$	$T - 92$ T - 92
$67$	$T + 796$ T + 796
$71$	$T + 426$ T + 426
$73$	$T + 1174$ T + 1174
$79$	$T - 842$ T - 842
$83$	$T + 852$ T + 852
$89$	$T - 1062$ T - 1062
$97$	$T + 1282$ T + 1282
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