LMFDB - Newform orbit 728.1.l.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [728,1,Mod(181,728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("728.181"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 728 = 2^{3} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	728.l (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$0.363319329197$
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 3.1.728.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.3709888.2
Stark unit:	Root of $x^{6} - 4250x^{5} - 559589x^{4} - 19190180x^{3} - 559589x^{2} - 4250x + 1$

$q$-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 $\left(\mathbb{Z}/728\mathbb{Z}\right)^\times$.

$n$	$183$	$365$	$521$	$561$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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} $

181.1

1.00000

−1.00000

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
728.l	odd	2	1	CM by $\Q(\sqrt{-182}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	728.1.l.d	yes	1
4.b	odd	2	1		2912.1.l.b		1
7.b	odd	2	1		728.1.l.c	yes	1
8.b	even	2	1		728.1.l.a	✓	1
8.d	odd	2	1		2912.1.l.d		1
13.b	even	2	1		728.1.l.b	yes	1
28.d	even	2	1		2912.1.l.c		1
52.b	odd	2	1		2912.1.l.a		1
56.e	even	2	1		2912.1.l.a		1
56.h	odd	2	1		728.1.l.b	yes	1
91.b	odd	2	1		728.1.l.a	✓	1
104.e	even	2	1		728.1.l.c	yes	1
104.h	odd	2	1		2912.1.l.c		1
364.h	even	2	1		2912.1.l.d		1
728.b	even	2	1		2912.1.l.b		1
728.l	odd	2	1	CM	728.1.l.d	yes	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
728.1.l.a	✓	1	8.b	even	2	1
728.1.l.a	✓	1	91.b	odd	2	1
728.1.l.b	yes	1	13.b	even	2	1
728.1.l.b	yes	1	56.h	odd	2	1
728.1.l.c	yes	1	7.b	odd	2	1
728.1.l.c	yes	1	104.e	even	2	1
728.1.l.d	yes	1	1.a	even	1	1	trivial
728.1.l.d	yes	1	728.l	odd	2	1	CM
2912.1.l.a		1	52.b	odd	2	1
2912.1.l.a		1	56.e	even	2	1
2912.1.l.b		1	4.b	odd	2	1
2912.1.l.b		1	728.b	even	2	1
2912.1.l.c		1	28.d	even	2	1
2912.1.l.c		1	104.h	odd	2	1
2912.1.l.d		1	8.d	odd	2	1
2912.1.l.d		1	364.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(728, [\chi])$:

$ T_{3} - 1 $

T3 - 1

$ T_{11} + 1 $

T11 + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T - 1 $ T - 1
$5$	$ T $ T
$7$	$ T + 1 $ T + 1
$11$	$ T + 1 $ T + 1
$13$	$ T + 1 $ T + 1
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T + 1 $ T + 1
$29$	$ T $ T
$31$	$ T - 1 $ T - 1
$37$	$ T + 1 $ T + 1
$41$	$ T - 1 $ T - 1
$43$	$ T $ T
$47$	$ T - 1 $ T - 1
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T - 1 $ T - 1
$67$	$ T + 1 $ T + 1
$71$	$ T $ T
$73$	$ T - 1 $ T - 1
$79$	$ T + 1 $ T + 1
$83$	$ T $ T
$89$	$ T + 2 $ T + 2
$97$	$ T - 1 $ T - 1
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Level:	\( N \)	\(=\)	\( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	728.l (of order \(2\), degree \(1\), minimal)

\(n\)	\(183\)	\(365\)	\(521\)	\(561\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-1\)

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