LMFDB - Newform orbit 243.1.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [243,1,Mod(242,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("243.242");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 243 = 3^{5} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	243.b (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:	$0.121272798070$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.243.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.243.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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} $

242.1

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	243.1.b.a	✓	1
3.b	odd	2	1	CM	243.1.b.a	✓	1
4.b	odd	2	1		3888.1.e.b		1
9.c	even	3	2		243.1.d.a		2
9.d	odd	6	2		243.1.d.a		2
12.b	even	2	1		3888.1.e.b		1
27.e	even	9	6		729.1.f.a		6
27.f	odd	18	6		729.1.f.a		6
36.f	odd	6	2		3888.1.q.b		2
36.h	even	6	2		3888.1.q.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
243.1.b.a	✓	1	1.a	even	1	1	trivial
243.1.b.a	✓	1	3.b	odd	2	1	CM
243.1.d.a		2	9.c	even	3	2
243.1.d.a		2	9.d	odd	6	2
729.1.f.a		6	27.e	even	9	6
729.1.f.a		6	27.f	odd	18	6
3888.1.e.b		1	4.b	odd	2	1
3888.1.e.b		1	12.b	even	2	1
3888.1.q.b		2	36.f	odd	6	2
3888.1.q.b		2	36.h	even	6	2

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(243, [\chi])$.

Hecke characteristic polynomials

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

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	243.b (of order \(2\), degree \(1\), minimal)

\(n\)	\(2\)
\(\chi(n)\)	\(-1\)

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