LMFDB - Newform orbit 2496.1.be.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,1,Mod(671,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 2, 1]))

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

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

chi := DirichletCharacter("2496.671");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$2496 = 2^{6} \cdot 3 \cdot 13$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2496.be (of order $4$ , 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:	$1.24566627153$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.421824.2

$q$ -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.

$f(q)$	$=$	$q - q^{3} + ( - i - 1) q^{7} + q^{9} + i q^{13} + (i + 1) q^{19} + (i + 1) q^{21} - i q^{25} - q^{27} + ( - i + 1) q^{31} + ( - i - 1) q^{37} - i q^{39} - 2 i q^{43} + i q^{49} + ( - i - 1) q^{57} - 2 i q^{61} + \cdots + (i + 1) q^{97} +O(q^{100})$ q - q^3 + (-z - 1) * q^7 + q^9 + z * q^13 + (z + 1) * q^19 + (z + 1) * q^21 - z * q^25 - q^27 + (-z + 1) * q^31 + (-z - 1) * q^37 - z * q^39 - 2z q^43 + z * q^49 + (-z - 1) * q^57 - 2z q^61 + (-z - 1) * q^63 + (z + 1) * q^67 + (-z + 1) * q^73 + z * q^75 + q^81 + (-z + 1) * q^91 + (z - 1) * q^93 + (z + 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{3} - 2 q^{7} + 2 q^{9} + 2 q^{19} + 2 q^{21} - 2 q^{27} + 2 q^{31} - 2 q^{37} - 2 q^{57} - 2 q^{63} + 2 q^{67} + 2 q^{73} + 2 q^{81} + 2 q^{91} - 2 q^{93} + 2 q^{97}+O(q^{100})$ 2 * q - 2 * q^3 - 2 * q^7 + 2 * q^9 + 2 * q^19 + 2 * q^21 - 2 * q^27 + 2 * q^31 - 2 * q^37 - 2 * q^57 - 2 * q^63 + 2 * q^67 + 2 * q^73 + 2 * q^81 + 2 * q^91 - 2 * q^93 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$ .

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$-1$	$i$	$-1$	$-1$

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}

671.1

		1.00000i
	−	1.00000i

−1.00000

−

1.00000i

1.00000

863.1

−1.00000

1.00000i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
104.m	even	4	1	inner
312.w	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2496.1.be.a	✓	2
3.b	odd	2	1	CM	2496.1.be.a	✓	2
4.b	odd	2	1		2496.1.be.d	yes	2
8.b	even	2	1		2496.1.be.c	yes	2
8.d	odd	2	1		2496.1.be.b	yes	2
12.b	even	2	1		2496.1.be.d	yes	2
13.d	odd	4	1		2496.1.be.b	yes	2
24.f	even	2	1		2496.1.be.b	yes	2
24.h	odd	2	1		2496.1.be.c	yes	2
39.f	even	4	1		2496.1.be.b	yes	2
52.f	even	4	1		2496.1.be.c	yes	2
104.j	odd	4	1		2496.1.be.d	yes	2
104.m	even	4	1	inner	2496.1.be.a	✓	2
156.l	odd	4	1		2496.1.be.c	yes	2
312.w	odd	4	1	inner	2496.1.be.a	✓	2
312.y	even	4	1		2496.1.be.d	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2496.1.be.a	✓	2	1.a	even	1	1	trivial
2496.1.be.a	✓	2	3.b	odd	2	1	CM
2496.1.be.a	✓	2	104.m	even	4	1	inner
2496.1.be.a	✓	2	312.w	odd	4	1	inner
2496.1.be.b	yes	2	8.d	odd	2	1
2496.1.be.b	yes	2	13.d	odd	4	1
2496.1.be.b	yes	2	24.f	even	2	1
2496.1.be.b	yes	2	39.f	even	4	1
2496.1.be.c	yes	2	8.b	even	2	1
2496.1.be.c	yes	2	24.h	odd	2	1
2496.1.be.c	yes	2	52.f	even	4	1
2496.1.be.c	yes	2	156.l	odd	4	1
2496.1.be.d	yes	2	4.b	odd	2	1
2496.1.be.d	yes	2	12.b	even	2	1
2496.1.be.d	yes	2	104.j	odd	4	1
2496.1.be.d	yes	2	312.y	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(2496, [\chi])$ :

T_{7}^{2} + 2T_{7} + 2

T_{19}^{2} - 2T_{19} + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$(T + 1)^{2}$ (T + 1)^2
$5$	$T^{2}$ T^2
$7$	$T^{2} + 2T + 2$ T^2 + 2*T + 2
$11$	$T^{2}$ T^2
$13$	$T^{2} + 1$ T^2 + 1
$17$	$T^{2}$ T^2
$19$	$T^{2} - 2T + 2$ T^2 - 2*T + 2
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} - 2T + 2$ T^2 - 2*T + 2
$37$	$T^{2} + 2T + 2$ T^2 + 2*T + 2
$41$	$T^{2}$ T^2
$43$	$T^{2} + 4$ T^2 + 4
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$T^{2} + 4$ T^2 + 4
$67$	$T^{2} - 2T + 2$ T^2 - 2*T + 2
$71$	$T^{2}$ T^2
$73$	$T^{2} - 2T + 2$ T^2 - 2*T + 2
$79$	$T^{2}$ T^2
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2} - 2T + 2$ T^2 - 2*T + 2
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