Compute spaces of half-integral weight modular forms#

Based on an algorithm in Basmaji’s thesis.

AUTHORS:

William Stein (2007-08)

sage.modular.modform.half_integral.half_integral_weight_modform_basis(chi, k, prec)[source]#

A basis for the space of weight \(k/2\) forms with character \(\chi\). The modulus of \(\chi\) must be divisible by \(16\) and \(k\) must be odd and \(>1\).

INPUT:

chi – a Dirichlet character with modulus divisible by 16
k – an odd integer > 1
prec – a positive integer

OUTPUT: a list of power series

Warning

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.
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.

EXAMPLES:

We compute some half-integral weight forms of level 16*7

Sage

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)]

Python

>>> from sage.all import *
>>> half_integral_weight_modform_basis(DirichletGroup(Integer(16)*Integer(7)).gen(0)**Integer(2),Integer(3),Integer(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

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)]

Python

>>> from sage.all import *
>>> half_integral_weight_modform_basis(DirichletGroup(Integer(16)*Integer(7)).gen(0)**Integer(2),Integer(3),Integer(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

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)]

Python

>>> from sage.all import *
>>> half_integral_weight_modform_basis(DirichletGroup(Integer(16),QQ).gen(1), Integer(3), Integer(10))
[]
>>> half_integral_weight_modform_basis(DirichletGroup(Integer(16),QQ).gen(1), Integer(5), Integer(10))
[q - 2*q^3 - 2*q^5 + 4*q^7 - q^9 + O(q^10)]
>>> half_integral_weight_modform_basis(DirichletGroup(Integer(16),QQ).gen(1), Integer(7), Integer(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)]
>>> half_integral_weight_modform_basis(DirichletGroup(Integer(16),QQ).gen(1), Integer(9), Integer(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)]

This example once raised an error (see Issue #5792).

Sage

sage: half_integral_weight_modform_basis(trivial_character(16),9,10)
[q - 2*q^2 + 4*q^3 - 8*q^4 + 4*q^6 - 16*q^7 + 48*q^8 - 15*q^9 + O(q^10),
 q^2 - 2*q^3 + 4*q^4 - 2*q^6 + 8*q^7 - 24*q^8 + O(q^10),
 q^3 - 2*q^4 - 4*q^7 + 12*q^8 + O(q^10),
 q^4 - 6*q^8 + O(q^10)]

Python

>>> from sage.all import *
>>> half_integral_weight_modform_basis(trivial_character(Integer(16)),Integer(9),Integer(10))
[q - 2*q^2 + 4*q^3 - 8*q^4 + 4*q^6 - 16*q^7 + 48*q^8 - 15*q^9 + O(q^10),
 q^2 - 2*q^3 + 4*q^4 - 2*q^6 + 8*q^7 - 24*q^8 + O(q^10),
 q^3 - 2*q^4 - 4*q^7 + 12*q^8 + O(q^10),
 q^4 - 6*q^8 + 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 \(S = S_{k+1}(\epsilon)\) be the space of cusp forms of even integer weight \(k+1\) and character \(\varepsilon = \chi \psi^{(k+1)/2}\), where \(\psi\) is the nontrivial mod-4 Dirichlet character. Let \(U\) be the subspace of \(S \times S\) of elements \((a,b)\) such that \(\Theta_2 a = \Theta_3 b\). Then \(U\) is isomorphic to \(S_{k/2}(\chi)\) via the map \((a,b) \mapsto a/\Theta_3\).