LMFDB - Newform orbit 3276.1.ek.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3276,1,Mod(649,3276)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3276, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5, 3]))

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

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

chi := DirichletCharacter("3276.649");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3276.ek (of order $6$ , degree $2$ , 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:	no
Analytic conductor:	$1.63493698139$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.15951590928.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

$f(q)$	$=$	$q + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{13} + \zeta_{6} q^{19} - \zeta_{6}^{2} q^{25} + \zeta_{6}^{2} q^{31} + (\zeta_{6}^{2} - 1) q^{37} + q^{43} + \zeta_{6}^{2} q^{49} + (\zeta_{6} + 1) q^{67} + \zeta_{6}^{2} q^{73} + \cdots + 2 q^{97} +O(q^{100})$ q + z * q^7 + z^2 * q^13 + z * q^19 - z^2 * q^25 + z^2 * q^31 + (z^2 - 1) * q^37 + q^43 + z^2 * q^49 + (z + 1) * q^67 + z^2 * q^73 - z * q^79 - q^91 + 2 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{7} - q^{13} + q^{19} + q^{25} - q^{31} - 3 q^{37} + 2 q^{43} - q^{49} + 3 q^{67} - q^{73} - q^{79} - 2 q^{91} + 4 q^{97}+O(q^{100})$ 2 * q + q^7 - q^13 + q^19 + q^25 - q^31 - 3 * q^37 + 2 * q^43 - q^49 + 3 * q^67 - q^73 - q^79 - 2 * q^91 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times$ .

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$-1$	$\zeta_{6}$	$1$

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}

649.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0.500000

−

0.866025i

1585.1

0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
91.s	odd	6	1	inner
273.ba	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3276.1.ek.b	yes	2
3.b	odd	2	1	CM	3276.1.ek.b	yes	2
7.d	odd	6	1		3276.1.ek.a	✓	2
13.b	even	2	1		3276.1.ek.a	✓	2
21.g	even	6	1		3276.1.ek.a	✓	2
39.d	odd	2	1		3276.1.ek.a	✓	2
91.s	odd	6	1	inner	3276.1.ek.b	yes	2
273.ba	even	6	1	inner	3276.1.ek.b	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3276.1.ek.a	✓	2	7.d	odd	6	1
3276.1.ek.a	✓	2	13.b	even	2	1
3276.1.ek.a	✓	2	21.g	even	6	1
3276.1.ek.a	✓	2	39.d	odd	2	1
3276.1.ek.b	yes	2	1.a	even	1	1	trivial
3276.1.ek.b	yes	2	3.b	odd	2	1	CM
3276.1.ek.b	yes	2	91.s	odd	6	1	inner
3276.1.ek.b	yes	2	273.ba	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{19}^{2} - T_{19} + 1$ T19^2 - T19 + 1 acting on $S_{1}^{\mathrm{new}}(3276, [\chi])$ .

Hecke characteristic polynomials

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