LMFDB - Newform orbit 256.6.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,6,Mod(1,256)] 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("256.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,0,0,-76,0,0,0,-243] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.0582578721$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 128)
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

−76.0000

−243.000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	256.6.a.b		1
4.b	odd	2	1	CM	256.6.a.b		1
8.b	even	2	1		256.6.a.c		1
8.d	odd	2	1		256.6.a.c		1
16.e	even	4	2		128.6.b.b	✓	2
16.f	odd	4	2		128.6.b.b	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
128.6.b.b	✓	2	16.e	even	4	2
128.6.b.b	✓	2	16.f	odd	4	2
256.6.a.b		1	1.a	even	1	1	trivial
256.6.a.b		1	4.b	odd	2	1	CM
256.6.a.c		1	8.b	even	2	1
256.6.a.c		1	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(256))$:

$ T_{3} $

T3

$ T_{5} + 76 $

T5 + 76

$ T_{7} $

T7

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T + 76 $ T + 76
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T - 244 $ T - 244
$17$	$ T + 2242 $ T + 2242
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T - 8564 $ T - 8564
$31$	$ T $ T
$37$	$ T + 11292 $ T + 11292
$41$	$ T - 20950 $ T - 20950
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T - 40244 $ T - 40244
$59$	$ T $ T
$61$	$ T - 54948 $ T - 54948
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T + 88806 $ T + 88806
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T - 51050 $ T - 51050
$97$	$ T - 92142 $ T - 92142
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Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

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