LMFDB - Newform orbit 680.1.k.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [680,1,Mod(339,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 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("680.339"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 680 = 2^{3} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	680.k (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:	$0.339364208590$
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 3.1.680.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.2.7860800.1
Stark unit:	Root of $x^{6} - 2486x^{5} + 233707x^{4} - 5717756x^{3} + 233707x^{2} - 2486x + 1$

$q$-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 $\left(\mathbb{Z}/680\mathbb{Z}\right)^\times$.

$n$	$137$	$241$	$341$	$511$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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} $

339.1

1.00000

−1.00000

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
680.k	odd	2	1	CM by $\Q(\sqrt{-170}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	680.1.k.d	yes	1
4.b	odd	2	1		2720.1.k.a		1
5.b	even	2	1		680.1.k.a	✓	1
5.c	odd	4	2		3400.1.g.e		2
8.b	even	2	1		2720.1.k.b		1
8.d	odd	2	1		680.1.k.b	yes	1
17.b	even	2	1		680.1.k.c	yes	1
20.d	odd	2	1		2720.1.k.c		1
40.e	odd	2	1		680.1.k.c	yes	1
40.f	even	2	1		2720.1.k.d		1
40.k	even	4	2		3400.1.g.d		2
68.d	odd	2	1		2720.1.k.d		1
85.c	even	2	1		680.1.k.b	yes	1
85.g	odd	4	2		3400.1.g.d		2
136.e	odd	2	1		680.1.k.a	✓	1
136.h	even	2	1		2720.1.k.c		1
340.d	odd	2	1		2720.1.k.b		1
680.h	even	2	1		2720.1.k.a		1
680.k	odd	2	1	CM	680.1.k.d	yes	1
680.u	even	4	2		3400.1.g.e		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
680.1.k.a	✓	1	5.b	even	2	1
680.1.k.a	✓	1	136.e	odd	2	1
680.1.k.b	yes	1	8.d	odd	2	1
680.1.k.b	yes	1	85.c	even	2	1
680.1.k.c	yes	1	17.b	even	2	1
680.1.k.c	yes	1	40.e	odd	2	1
680.1.k.d	yes	1	1.a	even	1	1	trivial
680.1.k.d	yes	1	680.k	odd	2	1	CM
2720.1.k.a		1	4.b	odd	2	1
2720.1.k.a		1	680.h	even	2	1
2720.1.k.b		1	8.b	even	2	1
2720.1.k.b		1	340.d	odd	2	1
2720.1.k.c		1	20.d	odd	2	1
2720.1.k.c		1	136.h	even	2	1
2720.1.k.d		1	40.f	even	2	1
2720.1.k.d		1	68.d	odd	2	1
3400.1.g.d		2	40.k	even	4	2
3400.1.g.d		2	85.g	odd	4	2
3400.1.g.e		2	5.c	odd	4	2
3400.1.g.e		2	680.u	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(680, [\chi])$:

$ T_{3} - 1 $

T3 - 1

$ T_{13} + 1 $

T13 + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T - 1 $ T - 1
$5$	$ T + 1 $ T + 1
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T + 1 $ T + 1
$17$	$ T + 1 $ T + 1
$19$	$ T + 1 $ T + 1
$23$	$ T $ T
$29$	$ T - 1 $ T - 1
$31$	$ T - 1 $ T - 1
$37$	$ T $ T
$41$	$ T $ T
$43$	$ T $ T
$47$	$ T + 1 $ T + 1
$53$	$ T + 1 $ T + 1
$59$	$ T + 1 $ T + 1
$61$	$ T - 1 $ T - 1
$67$	$ T $ T
$71$	$ T - 1 $ T - 1
$73$	$ T - 1 $ T - 1
$79$	$ T + 2 $ T + 2
$83$	$ T $ T
$89$	$ T + 1 $ T + 1
$97$	$ T - 1 $ T - 1
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Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	680.k (of order \(2\), degree \(1\), minimal)

\(n\)	\(137\)	\(241\)	\(341\)	\(511\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

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