LMFDB - Newform orbit 121.4.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,4,Mod(1,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	121.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:	$7.13923111069$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

8.00000

−8.00000

18.0000

37.0000

Atkin-Lehner signs

$ p $	Sign
$11$	$ +1 $

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	121.4.a.a	✓	1
3.b	odd	2	1		1089.4.a.f		1
4.b	odd	2	1		1936.4.a.a		1
11.b	odd	2	1	CM	121.4.a.a	✓	1
11.c	even	5	4		121.4.c.a		4
11.d	odd	10	4		121.4.c.a		4
33.d	even	2	1		1089.4.a.f		1
44.c	even	2	1		1936.4.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
121.4.a.a	✓	1	1.a	even	1	1	trivial
121.4.a.a	✓	1	11.b	odd	2	1	CM
121.4.c.a		4	11.c	even	5	4
121.4.c.a		4	11.d	odd	10	4
1089.4.a.f		1	3.b	odd	2	1
1089.4.a.f		1	33.d	even	2	1
1936.4.a.a		1	4.b	odd	2	1
1936.4.a.a		1	44.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} $ T2 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(121))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 8 $ T - 8
$5$	$ T - 18 $ T - 18
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T + 108 $ T + 108
$29$	$ T $ T
$31$	$ T - 340 $ T - 340
$37$	$ T + 434 $ T + 434
$41$	$ T $ T
$43$	$ T $ T
$47$	$ T + 36 $ T + 36
$53$	$ T + 738 $ T + 738
$59$	$ T + 720 $ T + 720
$61$	$ T $ T
$67$	$ T + 416 $ T + 416
$71$	$ T - 612 $ T - 612
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T - 1674 $ T - 1674
$97$	$ T + 34 $ T + 34
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
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