Half Integral Weight Forms

Basmaji’s 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}(\varepsilon) 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.

This algorithm is implemented in Sage. I’m sure it could be implemented in a way that is much faster than the current implementation...

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*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)]