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 :math:S = S_{k+1}(\varepsilon) be the space of cusp forms of even integer weight :math:k+1 and character :math:\varepsilon = \chi \psi^{(k+1)/2}, where :math:\psi is the nontrivial mod-4 Dirichlet character. Let :math:U be the subspace of :math:S \times S of elements :math:(a,b) such that :math:\Theta_2 a = \Theta_3 b. Then :math:U is isomorphic to :math:S_{k/2}(\chi) via the map :math:(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)]