LMFDB - Newform orbit 567.1.d.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,1,Mod(244,567)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("567.244");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$567 = 3^{4} \cdot 7$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	567.d (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$0.282969862163$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 3$ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.6751269.1

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of $\beta = \sqrt{3}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta q^{2} + 2 q^{4} - q^{7} - \beta q^{8} + \beta q^{11} + \beta q^{14} + q^{16} - 3 q^{22} + q^{25} - 2 q^{28} + q^{37} + q^{43} + 2 \beta q^{44} + q^{49} - \beta q^{50} + \beta q^{53} + \beta q^{56} + \cdots - \beta q^{98} +O(q^{100})$ q - b * q^2 + 2 * q^4 - q^7 - b * q^8 + b * q^11 + b * q^14 + q^16 - 3 * q^22 + q^25 - 2 * q^28 + q^37 + q^43 + 2b q^44 + q^49 - b * q^50 + b * q^53 + b * q^56 - q^64 - q^67 - b * q^71 - b * q^74 - b * q^77 - q^79 - b * q^86 - 3 * q^88 - b * q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{4} - 2 q^{7} + 2 q^{16} - 6 q^{22} + 2 q^{25} - 4 q^{28} + 2 q^{37} + 2 q^{43} + 2 q^{49} - 2 q^{64} - 2 q^{67} - 2 q^{79} - 6 q^{88}+O(q^{100})$ 2 * q + 4 * q^4 - 2 * q^7 + 2 * q^16 - 6 * q^22 + 2 * q^25 - 4 * q^28 + 2 * q^37 + 2 * q^43 + 2 * q^49 - 2 * q^64 - 2 * q^67 - 2 * q^79 - 6 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$ .

$n$	$325$	$407$
$\chi(n)$	$-1$	$1$

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}

244.1

1.73205
−1.73205

−1.73205

2.00000

−1.00000

−1.73205

244.2

1.73205

2.00000

−1.00000

1.73205

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	567.1.d.c	✓	2
3.b	odd	2	1	inner	567.1.d.c	✓	2
7.b	odd	2	1	CM	567.1.d.c	✓	2
7.c	even	3	2		3969.1.m.c		4
7.d	odd	6	2		3969.1.m.c		4
9.c	even	3	2		567.1.l.d		4
9.d	odd	6	2		567.1.l.d		4
21.c	even	2	1	inner	567.1.d.c	✓	2
21.g	even	6	2		3969.1.m.c		4
21.h	odd	6	2		3969.1.m.c		4
63.g	even	3	2		3969.1.k.e		4
63.h	even	3	2		3969.1.t.e		4
63.i	even	6	2		3969.1.t.e		4
63.j	odd	6	2		3969.1.t.e		4
63.k	odd	6	2		3969.1.k.e		4
63.l	odd	6	2		567.1.l.d		4
63.n	odd	6	2		3969.1.k.e		4
63.o	even	6	2		567.1.l.d		4
63.s	even	6	2		3969.1.k.e		4
63.t	odd	6	2		3969.1.t.e		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
567.1.d.c	✓	2	1.a	even	1	1	trivial
567.1.d.c	✓	2	3.b	odd	2	1	inner
567.1.d.c	✓	2	7.b	odd	2	1	CM
567.1.d.c	✓	2	21.c	even	2	1	inner
567.1.l.d		4	9.c	even	3	2
567.1.l.d		4	9.d	odd	6	2
567.1.l.d		4	63.l	odd	6	2
567.1.l.d		4	63.o	even	6	2
3969.1.k.e		4	63.g	even	3	2
3969.1.k.e		4	63.k	odd	6	2
3969.1.k.e		4	63.n	odd	6	2
3969.1.k.e		4	63.s	even	6	2
3969.1.m.c		4	7.c	even	3	2
3969.1.m.c		4	7.d	odd	6	2
3969.1.m.c		4	21.g	even	6	2
3969.1.m.c		4	21.h	odd	6	2
3969.1.t.e		4	63.h	even	3	2
3969.1.t.e		4	63.i	even	6	2
3969.1.t.e		4	63.j	odd	6	2
3969.1.t.e		4	63.t	odd	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} - 3$ T2^2 - 3 acting on $S_{1}^{\mathrm{new}}(567, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 3$ T^2 - 3
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$(T + 1)^{2}$ (T + 1)^2
$11$	$T^{2} - 3$ T^2 - 3
$13$	$T^{2}$ T^2
$17$	$T^{2}$ T^2
$19$	$T^{2}$ T^2
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2}$ T^2
$37$	$(T - 1)^{2}$ (T - 1)^2
$41$	$T^{2}$ T^2
$43$	$(T - 1)^{2}$ (T - 1)^2
$47$	$T^{2}$ T^2
$53$	$T^{2} - 3$ T^2 - 3
$59$	$T^{2}$ T^2
$61$	$T^{2}$ T^2
$67$	$(T + 1)^{2}$ (T + 1)^2
$71$	$T^{2} - 3$ T^2 - 3
$73$	$T^{2}$ T^2
$79$	$(T + 1)^{2}$ (T + 1)^2
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2}$ T^2
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