LMFDB - Newform orbit 288.3.e.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [288,3,Mod(161,288)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("288.161"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$N$	$=$	$288 = 2^{5} \cdot 3^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	288.e (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$7.84743161358$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 2$ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -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 $\beta = \sqrt{-2}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 7 \beta q^{5} - 24 q^{13} - 7 \beta q^{17} - 73 q^{25} + 41 \beta q^{29} + 70 q^{37} + 31 \beta q^{41} - 49 q^{49} + 17 \beta q^{53} - 22 q^{61} - 168 \beta q^{65} + 96 q^{73} + 98 q^{85} - 41 \beta q^{89} + \cdots + 144 q^{97} +O(q^{100})$ q + 7b q^5 - 24 * q^13 - 7b q^17 - 73 * q^25 + 41b q^29 + 70 * q^37 + 31b q^41 - 49 * q^49 + 17b q^53 - 22 * q^61 - 168b q^65 + 96 * q^73 + 98 * q^85 - 41b q^89 + 144 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 48 q^{13} - 146 q^{25} + 140 q^{37} - 98 q^{49} - 44 q^{61} + 192 q^{73} + 196 q^{85} + 288 q^{97}+O(q^{100})$ 2 * q - 48 * q^13 - 146 * q^25 + 140 * q^37 - 98 * q^49 - 44 * q^61 + 192 * q^73 + 196 * q^85 + 288 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$ .

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$1$

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}

161.1

	−	1.41421i
		1.41421i

−

9.89949i

161.2

9.89949i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.3.e.b	✓	2
3.b	odd	2	1	inner	288.3.e.b	✓	2
4.b	odd	2	1	CM	288.3.e.b	✓	2
8.b	even	2	1		576.3.e.e		2
8.d	odd	2	1		576.3.e.e		2
12.b	even	2	1	inner	288.3.e.b	✓	2
16.e	even	4	2		2304.3.h.e		4
16.f	odd	4	2		2304.3.h.e		4
24.f	even	2	1		576.3.e.e		2
24.h	odd	2	1		576.3.e.e		2
48.i	odd	4	2		2304.3.h.e		4
48.k	even	4	2		2304.3.h.e		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.3.e.b	✓	2	1.a	even	1	1	trivial
288.3.e.b	✓	2	3.b	odd	2	1	inner
288.3.e.b	✓	2	4.b	odd	2	1	CM
288.3.e.b	✓	2	12.b	even	2	1	inner
576.3.e.e		2	8.b	even	2	1
576.3.e.e		2	8.d	odd	2	1
576.3.e.e		2	24.f	even	2	1
576.3.e.e		2	24.h	odd	2	1
2304.3.h.e		4	16.e	even	4	2
2304.3.h.e		4	16.f	odd	4	2
2304.3.h.e		4	48.i	odd	4	2
2304.3.h.e		4	48.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(288, [\chi])$ :

T_{5}^{2} + 98

T5^2 + 98

T_{7}

T7

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} + 98$ T^2 + 98
$7$	$T^{2}$ T^2
$11$	$T^{2}$ T^2
$13$	$(T + 24)^{2}$ (T + 24)^2
$17$	$T^{2} + 98$ T^2 + 98
$19$	$T^{2}$ T^2
$23$	$T^{2}$ T^2
$29$	$T^{2} + 3362$ T^2 + 3362
$31$	$T^{2}$ T^2
$37$	$(T - 70)^{2}$ (T - 70)^2
$41$	$T^{2} + 1922$ T^2 + 1922
$43$	$T^{2}$ T^2
$47$	$T^{2}$ T^2
$53$	$T^{2} + 578$ T^2 + 578
$59$	$T^{2}$ T^2
$61$	$(T + 22)^{2}$ (T + 22)^2
$67$	$T^{2}$ T^2
$71$	$T^{2}$ T^2
$73$	$(T - 96)^{2}$ (T - 96)^2
$79$	$T^{2}$ T^2
$83$	$T^{2}$ T^2
$89$	$T^{2} + 3362$ T^2 + 3362
$97$	$(T - 144)^{2}$ (T - 144)^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 $\F_{q}$
Belyi maps

$n$ :		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