LMFDB - Newform orbit 3888.2.a.be

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3888,2,Mod(1,3888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$3888 = 2^{4} \cdot 3^{5}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,0,0,-6,0,0,0,0,0,6,0,0,0,0,0,-6,0,0,0,0,0,3,0,0,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.0458363059$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 3x - 1$ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$3$
Twist minimal:	no (minimal twist has level 486)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

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

$f(q)$	$=$	$q - \beta_1 q^{5} + ( - \beta_{2} - 2) q^{7} + (\beta_{2} - \beta_1) q^{11} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{13} + 2 \beta_{2} q^{17} + (2 \beta_1 - 2) q^{19} + (2 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{2} + \beta_1 + 1) q^{25}+ \cdots + ( - 2 \beta_{2} + 5 \beta_1 - 4) q^{97}+O(q^{100})$ q - b1 * q^5 + (-b2 - 2) * q^7 + (b2 - b1) * q^11 + (-2b2 + 2b1 + 2) * q^13 + 2b2 q^17 + (2b1 - 2) q^19 + (2b2 - 2b1) * q^23 + (b2 + b1 + 1) * q^25 + (2b1 - 3) q^29 + (b2 + b1 - 2) * q^31 + (-b2 + 4b1 + 3) q^35 + (2b2 - 2b1 + 2) * q^37 + (2b2 - 2b1 - 6) * q^41 - 2 * q^43 + (-4b2 + 2b1) * q^47 + (2b2 + b1 + 3) q^49 + (-3b2 + 2b1 - 6) * q^53 + (2b2 - b1 + 3) q^55 + (4b2 + 3) q^59 + (-2b2 - 2b1 - 4) * q^61 + (-4b2 - 6) q^65 + 4 * q^67 - 2b2 q^71 + (-b2 + 2b1 - 4) q^73 + (-b2 + 3b1 - 3) q^77 + (2b1 + 1) q^79 + (-b2 - 4b1) q^83 + (2b2 - 4b1 - 6) * q^85 + 2b2 q^89 + (-6b1 + 2) q^91 + (-2b2 - 12) q^95 + (-2b2 + 5b1 - 4) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 6 q^{7} + 6 q^{13} - 6 q^{19} + 3 q^{25} - 9 q^{29} - 6 q^{31} + 9 q^{35} + 6 q^{37} - 18 q^{41} - 6 q^{43} + 9 q^{49} - 18 q^{53} + 9 q^{55} + 9 q^{59} - 12 q^{61} - 18 q^{65} + 12 q^{67} - 12 q^{73}+ \cdots - 12 q^{97}+O(q^{100})$ 3 * q - 6 * q^7 + 6 * q^13 - 6 * q^19 + 3 * q^25 - 9 * q^29 - 6 * q^31 + 9 * q^35 + 6 * q^37 - 18 * q^41 - 6 * q^43 + 9 * q^49 - 18 * q^53 + 9 * q^55 + 9 * q^59 - 12 * q^61 - 18 * q^65 + 12 * q^67 - 12 * q^73 - 9 * q^77 + 3 * q^79 - 18 * q^85 + 6 * q^91 - 36 * q^95 - 12 * q^97

Basis of coefficient ring in terms of $\nu = \zeta_{18} + \zeta_{18}^{-1}$ :

$\beta_{1}$	$=$	$\nu^{2} + \nu - 2$ v^2 + v - 2
$\beta_{2}$	$=$	$-\nu^{2} + 2\nu + 2$ -v^2 + 2*v + 2

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

$\nu$	$=$	$( \beta_{2} + \beta_1 ) / 3$ (b2 + b1) / 3
$\nu^{2}$	$=$	$( -\beta_{2} + 2\beta _1 + 6 ) / 3$ (-b2 + 2*b1 + 6) / 3

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

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}

1.1

1.87939
−1.53209
−0.347296

−3.41147

−4.22668

1.2

1.18479

1.41147

1.3

2.22668

−3.18479

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3888.2.a.be		3
3.b	odd	2	1		3888.2.a.bf		3
4.b	odd	2	1		486.2.a.h	yes	3
12.b	even	2	1		486.2.a.g	✓	3
36.f	odd	6	2		486.2.c.g		6
36.h	even	6	2		486.2.c.h		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
486.2.a.g	✓	3	12.b	even	2	1
486.2.a.h	yes	3	4.b	odd	2	1
486.2.c.g		6	36.f	odd	6	2
486.2.c.h		6	36.h	even	6	2
3888.2.a.be		3	1.a	even	1	1	trivial
3888.2.a.bf		3	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3888))$ :

T_{5}^{3} - 9T_{5} + 9

T5^3 - 9*T5 + 9

T_{7}^{3} + 6T_{7}^{2} + 3T_{7} - 19

T7^3 + 6*T7^2 + 3*T7 - 19

T_{11}^{3} - 9T_{11} - 9

T11^3 - 9*T11 - 9

T_{13}^{3} - 6T_{13}^{2} - 24T_{13} + 136

T13^3 - 6*T13^2 - 24*T13 + 136

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3}$ T^3
$5$	$T^{3} - 9T + 9$ T^3 - 9*T + 9
$7$	$T^{3} + 6 T^{2} + \cdots - 19$ T^3 + 6T^2 + 3T - 19
$11$	$T^{3} - 9T - 9$ T^3 - 9*T - 9
$13$	$T^{3} - 6 T^{2} + \cdots + 136$ T^3 - 6T^2 - 24T + 136
$17$	$T^{3} - 36T + 72$ T^3 - 36*T + 72
$19$	$T^{3} + 6 T^{2} + \cdots - 136$ T^3 + 6T^2 - 24T - 136
$23$	$T^{3} - 36T - 72$ T^3 - 36*T - 72
$29$	$T^{3} + 9 T^{2} + \cdots - 153$ T^3 + 9T^2 - 9T - 153
$31$	$T^{3} + 6 T^{2} + \cdots - 73$ T^3 + 6T^2 - 15T - 73
$37$	$T^{3} - 6 T^{2} + \cdots - 8$ T^3 - 6T^2 - 24T - 8
$41$	$T^{3} + 18 T^{2} + \cdots - 72$ T^3 + 18T^2 + 72T - 72
$43$	$(T + 2)^{3}$ (T + 2)^3
$47$	$T^{3} - 108T - 216$ T^3 - 108*T - 216
$53$	$T^{3} + 18 T^{2} + \cdots - 153$ T^3 + 18T^2 + 45T - 153
$59$	$T^{3} - 9 T^{2} + \cdots + 981$ T^3 - 9T^2 - 117T + 981
$61$	$T^{3} + 12 T^{2} + \cdots - 152$ T^3 + 12T^2 - 60T - 152
$67$	$(T - 4)^{3}$ (T - 4)^3
$71$	$T^{3} - 36T - 72$ T^3 - 36*T - 72
$73$	$T^{3} + 12 T^{2} + \cdots - 17$ T^3 + 12T^2 + 21T - 17
$79$	$T^{3} - 3 T^{2} + \cdots - 37$ T^3 - 3T^2 - 33T - 37
$83$	$T^{3} - 189T + 999$ T^3 - 189*T + 999
$89$	$T^{3} - 36T + 72$ T^3 - 36*T + 72
$97$	$T^{3} + 12 T^{2} + \cdots - 467$ T^3 + 12T^2 - 123T - 467
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