LMFDB - Newform orbit 3328.2.b.y

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3328,2,Mod(1665,3328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3328, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3328.1665");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3328 = 2^{8} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3328.b (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:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 9x^{2} + 16$ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 a basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{3} + (2 \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} + 1) q^{7} + (\beta_{3} - 2) q^{9} + 2 \beta_1 q^{11} - \beta_{2} q^{13} + ( - \beta_{3} - 3) q^{15} + ( - 3 \beta_{3} + 1) q^{17} + 2 \beta_1 q^{19}+ \cdots + (8 \beta_{2} - 4 \beta_1) q^{99}+O(q^{100})$ q + b1 * q^3 + (2b2 + b1) q^5 + (-b3 + 1) * q^7 + (b3 - 2) * q^9 + 2b1 q^11 - b2 * q^13 + (-b3 - 3) * q^15 + (-3b3 + 1) q^17 + 2b1 q^19 + (-4b2 + b1) q^21 + 8 * q^23 - 3b3 q^25 + (4b2 + b1) q^27 + 2b2 q^29 + 4 * q^31 + (2b3 - 10) q^33 + (-4b2 - b1) q^35 + (2b2 - 3b1) * q^37 + (b3 - 1) * q^39 - 2b3 q^41 + (8b2 + b1) q^43 + 2b2 q^45 + (-3b3 - 5) q^47 + (-b3 - 2) * q^49 + (-12b2 + b1) q^51 + (-2b2 - 2b1) * q^53 + (-2b3 - 6) q^55 + (2b3 - 10) q^57 - 2b1 q^59 + (-6b2 + 2b1) * q^61 + (2b3 - 6) q^63 + (b3 + 1) * q^65 - 2b1 q^67 + 8b1 q^69 + (-3b3 + 3) q^71 + 6 * q^73 - 12b2 q^75 + (-8b2 + 2b1) * q^77 + 8 * q^79 - 7 * q^81 + (8b2 + 4b1) * q^83 + (-16b2 - 5b1) * q^85 + (-2b3 + 2) q^87 - 10 * q^89 + b1 * q^91 + 4b1 q^93 + (-2b3 - 6) q^95 + (4b3 - 2) q^97 + (8b2 - 4b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{7} - 6 q^{9} - 14 q^{15} - 2 q^{17} + 32 q^{23} - 6 q^{25} + 16 q^{31} - 36 q^{33} - 2 q^{39} - 4 q^{41} - 26 q^{47} - 10 q^{49} - 28 q^{55} - 36 q^{57} - 20 q^{63} + 6 q^{65} + 6 q^{71} + 24 q^{73}+ \cdots - 28 q^{95}+O(q^{100})$ 4 * q + 2 * q^7 - 6 * q^9 - 14 * q^15 - 2 * q^17 + 32 * q^23 - 6 * q^25 + 16 * q^31 - 36 * q^33 - 2 * q^39 - 4 * q^41 - 26 * q^47 - 10 * q^49 - 28 * q^55 - 36 * q^57 - 20 * q^63 + 6 * q^65 + 6 * q^71 + 24 * q^73 + 32 * q^79 - 28 * q^81 + 4 * q^87 - 40 * q^89 - 28 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} + 9x^{2} + 16$ x^4 + 9*x^2 + 16 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{3} + 5\nu ) / 4$ (v^3 + 5*v) / 4
$\beta_{3}$	$=$	$\nu^{2} + 5$ v^2 + 5

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{3} - 5$ b3 - 5
$\nu^{3}$	$=$	$4\beta_{2} - 5\beta_1$ 4b2 - 5b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$ .

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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}

1665.1

	−	2.56155i
	−	1.56155i
		1.56155i
		2.56155i

−

2.56155i

−

0.561553i

2.56155

−3.56155

1665.2

−

1.56155i

−

3.56155i

−1.56155

0.561553

1665.3

1.56155i

3.56155i

−1.56155

0.561553

1665.4

2.56155i

0.561553i

2.56155

−3.56155

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3328.2.b.y		4
4.b	odd	2	1		3328.2.b.w		4
8.b	even	2	1	inner	3328.2.b.y		4
8.d	odd	2	1		3328.2.b.w		4
16.e	even	4	1		104.2.a.b	✓	2
16.e	even	4	1		832.2.a.k		2
16.f	odd	4	1		208.2.a.e		2
16.f	odd	4	1		832.2.a.n		2
48.i	odd	4	1		936.2.a.j		2
48.i	odd	4	1		7488.2.a.cu		2
48.k	even	4	1		1872.2.a.u		2
48.k	even	4	1		7488.2.a.cv		2
80.i	odd	4	1		2600.2.d.k		4
80.k	odd	4	1		5200.2.a.bw		2
80.q	even	4	1		2600.2.a.p		2
80.t	odd	4	1		2600.2.d.k		4
112.l	odd	4	1		5096.2.a.m		2
208.l	even	4	1		2704.2.f.k		4
208.m	odd	4	1		1352.2.f.c		4
208.o	odd	4	1		2704.2.a.p		2
208.p	even	4	1		1352.2.a.g		2
208.r	odd	4	1		1352.2.f.c		4
208.s	even	4	1		2704.2.f.k		4
208.be	odd	12	2		1352.2.o.d		8
208.bh	even	12	2		1352.2.i.d		4
208.bj	even	12	2		1352.2.i.f		4
208.bl	odd	12	2		1352.2.o.d		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.a.b	✓	2	16.e	even	4	1
208.2.a.e		2	16.f	odd	4	1
832.2.a.k		2	16.e	even	4	1
832.2.a.n		2	16.f	odd	4	1
936.2.a.j		2	48.i	odd	4	1
1352.2.a.g		2	208.p	even	4	1
1352.2.f.c		4	208.m	odd	4	1
1352.2.f.c		4	208.r	odd	4	1
1352.2.i.d		4	208.bh	even	12	2
1352.2.i.f		4	208.bj	even	12	2
1352.2.o.d		8	208.be	odd	12	2
1352.2.o.d		8	208.bl	odd	12	2
1872.2.a.u		2	48.k	even	4	1
2600.2.a.p		2	80.q	even	4	1
2600.2.d.k		4	80.i	odd	4	1
2600.2.d.k		4	80.t	odd	4	1
2704.2.a.p		2	208.o	odd	4	1
2704.2.f.k		4	208.l	even	4	1
2704.2.f.k		4	208.s	even	4	1
3328.2.b.w		4	4.b	odd	2	1
3328.2.b.w		4	8.d	odd	2	1
3328.2.b.y		4	1.a	even	1	1	trivial
3328.2.b.y		4	8.b	even	2	1	inner
5096.2.a.m		2	112.l	odd	4	1
5200.2.a.bw		2	80.k	odd	4	1
7488.2.a.cu		2	48.i	odd	4	1
7488.2.a.cv		2	48.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3328, [\chi])$ :

T_{3}^{4} + 9T_{3}^{2} + 16

T_{5}^{4} + 13T_{5}^{2} + 4

T_{7}^{2} - T_{7} - 4

T_{11}^{4} + 36T_{11}^{2} + 256

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} + 9T^{2} + 16$ T^4 + 9*T^2 + 16
$5$	$T^{4} + 13T^{2} + 4$ T^4 + 13*T^2 + 4
$7$	$(T^{2} - T - 4)^{2}$ (T^2 - T - 4)^2
$11$	$T^{4} + 36T^{2} + 256$ T^4 + 36*T^2 + 256
$13$	$(T^{2} + 1)^{2}$ (T^2 + 1)^2
$17$	$(T^{2} + T - 38)^{2}$ (T^2 + T - 38)^2
$19$	$T^{4} + 36T^{2} + 256$ T^4 + 36*T^2 + 256
$23$	$(T - 8)^{4}$ (T - 8)^4
$29$	$(T^{2} + 4)^{2}$ (T^2 + 4)^2
$31$	$(T - 4)^{4}$ (T - 4)^4
$37$	$T^{4} + 101T^{2} + 676$ T^4 + 101*T^2 + 676
$41$	$(T^{2} + 2 T - 16)^{2}$ (T^2 + 2*T - 16)^2
$43$	$T^{4} + 121T^{2} + 2704$ T^4 + 121*T^2 + 2704
$47$	$(T^{2} + 13 T + 4)^{2}$ (T^2 + 13*T + 4)^2
$53$	$T^{4} + 36T^{2} + 256$ T^4 + 36*T^2 + 256
$59$	$T^{4} + 36T^{2} + 256$ T^4 + 36*T^2 + 256
$61$	$T^{4} + 132T^{2} + 1024$ T^4 + 132*T^2 + 1024
$67$	$T^{4} + 36T^{2} + 256$ T^4 + 36*T^2 + 256
$71$	$(T^{2} - 3 T - 36)^{2}$ (T^2 - 3*T - 36)^2
$73$	$(T - 6)^{4}$ (T - 6)^4
$79$	$(T - 8)^{4}$ (T - 8)^4
$83$	$T^{4} + 208T^{2} + 1024$ T^4 + 208*T^2 + 1024
$89$	$(T + 10)^{4}$ (T + 10)^4
$97$	$(T^{2} - 68)^{2}$ (T^2 - 68)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more