LMFDB - Newform orbit 576.5.g.o

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,5,Mod(127,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("576.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$N$	$=$	$576 = 2^{6} \cdot 3^{2}$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	576.g (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:	$59.5410987363$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{2} + 1$ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{8}\cdot 3^{2}$
Twist minimal:	no (minimal twist has level 96)
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.

$f(q)$	$=$	$q + (\beta_{2} + 22) q^{5} + (\beta_{3} - \beta_1) q^{7} + (\beta_{3} - 22 \beta_1) q^{11} + ( - 2 \beta_{2} + 14) q^{13} + ( - 22 \beta_{2} + 6) q^{17} + ( - 23 \beta_{3} - 18 \beta_1) q^{19} + ( - 24 \beta_{3} - 94 \beta_1) q^{23}+ \cdots + ( - 252 \beta_{2} - 2942) q^{97}+O(q^{100})$ q + (b2 + 22) * q^5 + (b3 - b1) * q^7 + (b3 - 22b1) q^11 + (-2b2 + 14) q^13 + (-22b2 + 6) q^17 + (-23b3 - 18b1) * q^19 + (-24b3 - 94b1) * q^23 + (44b2 + 291) q^25 + (-25b2 - 418) q^29 + (-5b3 - 361b1) * q^31 + (18b3 + 86b1) * q^35 + (52b2 - 1578) q^37 + (-26b2 + 1126) q^41 + (43b3 + 178b1) * q^43 + (50b3 - 418b1) * q^47 + (8b2 + 1953) q^49 + (-145b2 - 834) q^53 + (-66b3 - 376b1) * q^55 + (-157b3 - 280b1) * q^59 + (240b2 - 250) q^61 + (-30b2 - 556) q^65 + (103b3 + 1240b1) * q^67 + (204b3 - 950b1) * q^71 + (8b2 + 4338) q^73 + (92b2 - 784) q^77 + (291b3 - 773b1) * q^79 + (311b3 - 950b1) * q^83 + (-478b2 - 9372) q^85 + (140b2 + 5214) q^89 + (22b3 - 230b1) * q^91 + (-578b3 - 2880b1) * q^95 + (-252b2 - 2942) q^97
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 88 q^{5} + 56 q^{13} + 24 q^{17} + 1164 q^{25} - 1672 q^{29} - 6312 q^{37} + 4504 q^{41} + 7812 q^{49} - 3336 q^{53} - 1000 q^{61} - 2224 q^{65} + 17352 q^{73} - 3136 q^{77} - 37488 q^{85} + 20856 q^{89}+ \cdots - 11768 q^{97}+O(q^{100})$ 4 * q + 88 * q^5 + 56 * q^13 + 24 * q^17 + 1164 * q^25 - 1672 * q^29 - 6312 * q^37 + 4504 * q^41 + 7812 * q^49 - 3336 * q^53 - 1000 * q^61 - 2224 * q^65 + 17352 * q^73 - 3136 * q^77 - 37488 * q^85 + 20856 * q^89 - 11768 * q^97

Basis of coefficient ring

$\beta_{1}$	$=$	$4\zeta_{12}^{3}$ 4*v^3
$\beta_{2}$	$=$	$-12\zeta_{12}^{3} + 24\zeta_{12}$ -12v^3 + 24v
$\beta_{3}$	$=$	$24\zeta_{12}^{2} - 12$ 24*v^2 - 12

Display $\zeta_{12}^j$ in terms of $\beta_i$

$\zeta_{12}$	$=$	$( \beta_{2} + 3\beta_1 ) / 24$ (b2 + 3*b1) / 24
$\zeta_{12}^{2}$	$=$	$( \beta_{3} + 12 ) / 24$ (b3 + 12) / 24
$\zeta_{12}^{3}$	$=$	$( \beta_1 ) / 4$ (b1) / 4

Display $\beta_i$ in terms of $\zeta_{12}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$ .

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$1$

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}

127.1

−0.866025	+	0.500000i
−0.866025	−	0.500000i
0.866025	−	0.500000i
0.866025	+	0.500000i

1.21539

−

24.7846i

127.2

1.21539

24.7846i

127.3

42.7846

−

16.7846i

127.4

42.7846

16.7846i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	576.5.g.o		4
3.b	odd	2	1		192.5.g.c		4
4.b	odd	2	1	inner	576.5.g.o		4
8.b	even	2	1		288.5.g.c		4
8.d	odd	2	1		288.5.g.c		4
12.b	even	2	1		192.5.g.c		4
24.f	even	2	1		96.5.g.b	✓	4
24.h	odd	2	1		96.5.g.b	✓	4
48.i	odd	4	1		768.5.b.a		4
48.i	odd	4	1		768.5.b.f		4
48.k	even	4	1		768.5.b.a		4
48.k	even	4	1		768.5.b.f		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.5.g.b	✓	4	24.f	even	2	1
96.5.g.b	✓	4	24.h	odd	2	1
192.5.g.c		4	3.b	odd	2	1
192.5.g.c		4	12.b	even	2	1
288.5.g.c		4	8.b	even	2	1
288.5.g.c		4	8.d	odd	2	1
576.5.g.o		4	1.a	even	1	1	trivial
576.5.g.o		4	4.b	odd	2	1	inner
768.5.b.a		4	48.i	odd	4	1
768.5.b.a		4	48.k	even	4	1
768.5.b.f		4	48.i	odd	4	1
768.5.b.f		4	48.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} - 44T_{5} + 52$ T5^2 - 44*T5 + 52 acting on $S_{5}^{\mathrm{new}}(576, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4}$ T^4
$5$	$(T^{2} - 44 T + 52)^{2}$ (T^2 - 44*T + 52)^2
$7$	$T^{4} + 896 T^{2} + 173056$ T^4 + 896*T^2 + 173056
$11$	$T^{4} + 16352 T^{2} + 53465344$ T^4 + 16352*T^2 + 53465344
$13$	$(T^{2} - 28 T - 1532)^{2}$ (T^2 - 28*T - 1532)^2
$17$	$(T^{2} - 12 T - 209052)^{2}$ (T^2 - 12*T - 209052)^2
$19$	$T^{4} + \cdots + 49882542336$ T^4 + 467424*T^2 + 49882542336
$23$	$T^{4} + \cdots + 11546791936$ T^4 + 780416*T^2 + 11546791936
$29$	$(T^{2} + 836 T - 95276)^{2}$ (T^2 + 836*T - 95276)^2
$31$	$T^{4} + \cdots + 4302869840896$ T^4 + 4191872*T^2 + 4302869840896
$37$	$(T^{2} + 3156 T + 1321956)^{2}$ (T^2 + 3156*T + 1321956)^2
$41$	$(T^{2} - 2252 T + 975844)^{2}$ (T^2 - 2252*T + 975844)^2
$43$	$T^{4} + \cdots + 85161246976$ T^4 + 2611424*T^2 + 85161246976
$47$	$T^{4} + \cdots + 2943228461056$ T^4 + 7751168*T^2 + 2943228461056
$53$	$(T^{2} + 1668 T - 8387244)^{2}$ (T^2 + 1668*T - 8387244)^2
$59$	$T^{4} + \cdots + 88246634785024$ T^4 + 23805536*T^2 + 88246634785024
$61$	$(T^{2} + 500 T - 24820700)^{2}$ (T^2 + 500*T - 24820700)^2
$67$	$T^{4} + \cdots + 400740822694144$ T^4 + 58369376*T^2 + 400740822694144
$71$	$T^{4} + \cdots + 12518236524544$ T^4 + 64836224*T^2 + 12518236524544
$73$	$(T^{2} - 8676 T + 18790596)^{2}$ (T^2 - 8676*T + 18790596)^2
$79$	$T^{4} + \cdots + 730173784105984$ T^4 + 92285312*T^2 + 730173784105984
$83$	$T^{4} + \cdots + 747665461014784$ T^4 + 112446944*T^2 + 747665461014784
$89$	$(T^{2} - 10428 T + 18718596)^{2}$ (T^2 - 10428*T + 18718596)^2
$97$	$(T^{2} + 5884 T - 18778364)^{2}$ (T^2 + 5884*T - 18778364)^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