LMFDB - Newform orbit 2016.1.l.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,1,Mod(433,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("2016.433");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$2016 = 2^{5} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2016.l (of order $2$ , degree $1$ , not 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:	$1.00611506547$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 504)
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{-6}, \sqrt{-7})$
Artin image:	$D_4:C_2$
Artin field:	Galois closure of 8.0.1792336896.7

$q$ -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.

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$ .

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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}

433.1

		1.00000i
	−	1.00000i

−1.00000

433.2

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7})$
24.h	odd	2	1	CM by $\Q(\sqrt{-6})$
168.i	even	2	1	RM by $\Q(\sqrt{42})$
3.b	odd	2	1	inner
8.b	even	2	1	inner
21.c	even	2	1	inner
56.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2016.1.l.b		2
3.b	odd	2	1	inner	2016.1.l.b		2
4.b	odd	2	1		504.1.l.b	✓	2
7.b	odd	2	1	CM	2016.1.l.b		2
8.b	even	2	1	inner	2016.1.l.b		2
8.d	odd	2	1		504.1.l.b	✓	2
12.b	even	2	1		504.1.l.b	✓	2
21.c	even	2	1	inner	2016.1.l.b		2
24.f	even	2	1		504.1.l.b	✓	2
24.h	odd	2	1	CM	2016.1.l.b		2
28.d	even	2	1		504.1.l.b	✓	2
28.f	even	6	2		3528.1.bw.b		4
28.g	odd	6	2		3528.1.bw.b		4
56.e	even	2	1		504.1.l.b	✓	2
56.h	odd	2	1	inner	2016.1.l.b		2
56.k	odd	6	2		3528.1.bw.b		4
56.m	even	6	2		3528.1.bw.b		4
84.h	odd	2	1		504.1.l.b	✓	2
84.j	odd	6	2		3528.1.bw.b		4
84.n	even	6	2		3528.1.bw.b		4
168.e	odd	2	1		504.1.l.b	✓	2
168.i	even	2	1	RM	2016.1.l.b		2
168.v	even	6	2		3528.1.bw.b		4
168.be	odd	6	2		3528.1.bw.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.1.l.b	✓	2	4.b	odd	2	1
504.1.l.b	✓	2	8.d	odd	2	1
504.1.l.b	✓	2	12.b	even	2	1
504.1.l.b	✓	2	24.f	even	2	1
504.1.l.b	✓	2	28.d	even	2	1
504.1.l.b	✓	2	56.e	even	2	1
504.1.l.b	✓	2	84.h	odd	2	1
504.1.l.b	✓	2	168.e	odd	2	1
2016.1.l.b		2	1.a	even	1	1	trivial
2016.1.l.b		2	3.b	odd	2	1	inner
2016.1.l.b		2	7.b	odd	2	1	CM
2016.1.l.b		2	8.b	even	2	1	inner
2016.1.l.b		2	21.c	even	2	1	inner
2016.1.l.b		2	24.h	odd	2	1	CM
2016.1.l.b		2	56.h	odd	2	1	inner
2016.1.l.b		2	168.i	even	2	1	RM
3528.1.bw.b		4	28.f	even	6	2
3528.1.bw.b		4	28.g	odd	6	2
3528.1.bw.b		4	56.k	odd	6	2
3528.1.bw.b		4	56.m	even	6	2
3528.1.bw.b		4	84.j	odd	6	2
3528.1.bw.b		4	84.n	even	6	2
3528.1.bw.b		4	168.v	even	6	2
3528.1.bw.b		4	168.be	odd	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{11}^{2} + 4$ T11^2 + 4 acting on $S_{1}^{\mathrm{new}}(2016, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$(T + 1)^{2}$ (T + 1)^2
$11$	$T^{2} + 4$ T^2 + 4
$13$	$T^{2}$ T^2
$17$	$T^{2}$ T^2
$19$	$T^{2}$ T^2
$23$	$T^{2}$ T^2
$29$	$T^{2} + 4$ T^2 + 4
$31$	$T^{2}$ T^2
$37$	$T^{2}$ T^2
$41$	$T^{2}$ T^2
$43$	$T^{2}$ T^2
$47$	$T^{2}$ T^2
$53$	$T^{2} + 4$ T^2 + 4
$59$	$T^{2}$ T^2
$61$	$T^{2}$ T^2
$67$	$T^{2}$ T^2
$71$	$T^{2}$ T^2
$73$	$T^{2}$ T^2
$79$	$(T - 2)^{2}$ (T - 2)^2
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2}$ T^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q - q^{7} - 2 i q^{11} - q^{25} - 2 i q^{29} + q^{49} - 2 i q^{53} + 2 i q^{77} + 2 q^{79} +O(q^{100})$ q - q^7 - 2z q^11 - q^25 - 2z q^29 + q^49 - 2z q^53 + 2z q^77 + 2 * q^79
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{7} - 2 q^{25} + 2 q^{49} + 4 q^{79}+O(q^{100})$ 2 * q - 2 * q^7 - 2 * q^25 + 2 * q^49 + 4 * q^79

$n$ :		e.g. 2-40 or 990-1000
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