Module: sage.modular.modform.half_integral
Compute spaces of half-integral weight modular forms.
Based on an algorithm in Basmaji's thesis.
Author: William Stein, 2007-08
Module-level Functions
chi, k, prec) |
A basis for the space of weight
forms with character
.
The modulus of
must be divisible by
and
must be odd
and
.
Input:
WARNINGS:
(1) This code is very slow because it requests computation of a basis of modular forms for integral weight spaces, and that computation is still very slow.
(2) If you give an input prec that is too small, then the output list of power series may be larger than the dimension of the space of half-integral forms.
We compute some half-integral weight forms of level 16*7
sage: half_integral_weight_modform_basis(DirichletGroup(16*7).0^2,3,30) [q - 2*q^2 - q^9 + 2*q^14 + 6*q^18 - 2*q^21 - 4*q^22 - q^25 + O(q^30), q^2 - q^14 - 3*q^18 + 2*q^22 + O(q^30), q^4 - q^8 - q^16 + q^28 + O(q^30), q^7 - 2*q^15 + O(q^30)]
The following illustrates that choosing too low of a precision can give an incorrect answer.
sage: half_integral_weight_modform_basis(DirichletGroup(16*7).0^2,3,20) [q - 2*q^2 - q^9 + 2*q^14 + 6*q^18 + O(q^20), q^2 - q^14 - 3*q^18 + O(q^20), q^4 - 2*q^8 + 2*q^12 - 4*q^16 + O(q^20), q^7 - 2*q^8 + 4*q^12 - 2*q^15 - 6*q^16 + O(q^20), q^8 - 2*q^12 + 3*q^16 + O(q^20)]
We compute some spaces of low level and the first few possible weights.
sage: half_integral_weight_modform_basis(DirichletGroup(16,QQ).1, 3, 10) [] sage: half_integral_weight_modform_basis(DirichletGroup(16,QQ).1, 5, 10) [q - 2*q^3 - 2*q^5 + 4*q^7 - q^9 + O(q^10)] sage: half_integral_weight_modform_basis(DirichletGroup(16,QQ).1, 7, 10) [q - 2*q^2 + 4*q^3 + 4*q^4 - 10*q^5 - 16*q^7 + 19*q^9 + O(q^10), q^2 - 2*q^3 - 2*q^4 + 4*q^5 + 4*q^7 - 8*q^9 + O(q^10), q^3 - 2*q^5 - 2*q^7 + 4*q^9 + O(q^10)] sage: half_integral_weight_modform_basis(DirichletGroup(16,QQ).1, 9, 10) [q - 2*q^2 + 4*q^3 - 8*q^4 + 14*q^5 + 16*q^6 - 40*q^7 + 16*q^8 - 57*q^9 + O(q^10), q^2 - 2*q^3 + 4*q^4 - 8*q^5 - 8*q^6 + 20*q^7 - 8*q^8 + 32*q^9 + O(q^10), q^3 - 2*q^4 + 4*q^5 + 4*q^6 - 10*q^7 - 16*q^9 + O(q^10), q^4 - 2*q^5 - 2*q^6 + 4*q^7 + 4*q^9 + O(q^10), q^5 - 2*q^7 - 2*q^9 + O(q^10)]
ALGORITHM: Basmaji (page 55 of his Essen thesis, "Ein Algorithmus zur Berechnung von Hecke-Operatoren und Anwendungen auf modulare Kurven", http://wstein.org/scans/papers/basmaji/).
Let
be the space of cusp forms of even
integer weight
and character
,
where
is the nontrivial mod-4 Dirichlet character. Let
be the subspace of
of elements
such that
. Then
is isomorphic to
via the map
.
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