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– positive integer
 - OUTPUT: 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: 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)] - >>> 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: 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)] - >>> 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: 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)] - >>> 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: 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)] - >>> 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 \((a,b) \mapsto a/\Theta_3\).