LMFDB - Newform orbit 441.8.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [441,8,Mod(1,441)] 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("441.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,13,0,41,0,0,0,-1131,0,0,-8684,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$137.761796238$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 49)
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

13.0000

41.0000

−1131.00

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$7$	$ -1 $

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.8.a.f		1
3.b	odd	2	1		49.8.a.a	✓	1
7.b	odd	2	1	CM	441.8.a.f		1
21.c	even	2	1		49.8.a.a	✓	1
21.g	even	6	2		49.8.c.c		2
21.h	odd	6	2		49.8.c.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
49.8.a.a	✓	1	3.b	odd	2	1
49.8.a.a	✓	1	21.c	even	2	1
49.8.c.c		2	21.g	even	6	2
49.8.c.c		2	21.h	odd	6	2
441.8.a.f		1	1.a	even	1	1	trivial
441.8.a.f		1	7.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 $S_{8}^{\mathrm{new}}(\Gamma_0(441))$:

$ T_{2} - 13 $

T2 - 13

$ T_{5} $

T5

$ T_{13} $

T13

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 13 $ T - 13
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T + 8684 $ T + 8684
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T - 67976 $ T - 67976
$29$	$ T - 253622 $ T - 253622
$31$	$ T $ T
$37$	$ T + 278382 $ T + 278382
$41$	$ T $ T
$43$	$ T - 1014204 $ T - 1014204
$47$	$ T $ T
$53$	$ T - 1804610 $ T - 1804610
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T + 4863092 $ T + 4863092
$71$	$ T - 3721744 $ T - 3721744
$73$	$ T $ T
$79$	$ T - 1101752 $ T - 1101752
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T $ T
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Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(n\):		e.g. 2-40 or 80-90
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