LMFDB - Newform orbit 243.9.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [243,9,Mod(242,243)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("243.242"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 243 = 3^{5} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	243.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,0,0,256,0,0,-4273] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$98.9930022449$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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

256.000

−4273.00

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	243.9.b.a	✓	1
3.b	odd	2	1	CM	243.9.b.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
243.9.b.a	✓	1	1.a	even	1	1	trivial
243.9.b.a	✓	1	3.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{9}^{\mathrm{new}}(243, [\chi])$:

$ T_{2} $

T2

$ T_{7} + 4273 $

T7 + 4273

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 4273 $ T + 4273
$11$	$ T $ T
$13$	$ T + 35806 $ T + 35806
$17$	$ T $ T
$19$	$ T + 258526 $ T + 258526
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T - 1225967 $ T - 1225967
$37$	$ T - 2964959 $ T - 2964959
$41$	$ T $ T
$43$	$ T - 3344879 $ T - 3344879
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T - 24133919 $ T - 24133919
$67$	$ T + 31874833 $ T + 31874833
$71$	$ T $ T
$73$	$ T + 55236481 $ T + 55236481
$79$	$ T + 18887038 $ T + 18887038
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T - 176908034 $ T - 176908034
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	243.b (of order \(2\), degree \(1\), minimal)

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

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