/hecke_mf

Hecke modular forms implementation for sage

Primary LanguagePython

Hecke modular forms

An implementation for sage:

  • Moved to sage: http://trac.sagemath.org/ticket/16134

  • At the moment: Only support for forms with respect to the full Hecke triangle group for n=3, 4, 5, ...

  • The ring of modular forms as a commutative algebra.

  • The space of modular forms of given weight as a module.

  • Supported analytic types (implemented as an extended FiniteLatticePoset class):

    • meromorphic
    • weakly holomorphic
    • holomorphic
    • cuspidal
    • Support for quasi modular forms for all of the above types

  • Exact calculations (no precision argument is required).

    The calculations are based on the three generators of the graded algebra: x=f_rho, y=f_i, z=E2. Every form has a representation as a rational function in x, y, z.

    Checks are performed to determine the analytic type of elements.

  • Fourier expansion with (exact) coefficients in Frac(R)[d], where R is some base ring (e.g. ZZ) and d is a formal parameter corresponding to a (possibly) transcendental number which turns up in the Fourier expansion.

    It is also possible to evaluate d numerically.

    The Fourier expansion is (should be) determined exactly with the specified precision.

  • For arithmetic groups the d is calculated exactly.

  • Evaluation of elements, viewed as functions from the upper half plane. This uses the modularity properties for faster/more precise evaluation. However the precision of the result depends on the precision specified for the Fourier expansion.

  • Calculation of derivatives and serre derivatives.

  • Basis for weakly holomorphic modular forms.

  • Faber polynomials.

  • (Exactly) determine weakly holomorphic modular forms by their initial Fourier coefficients.

  • Dimension and basis for holomorphic or cuspidal (quasi) modular forms.

  • Coordinate vectors for holomorphic modular forms and cusp forms.

  • Subspaces (with respect to a basis) for ambient spaces that support coordinate vectors, together with coordinate vectors for subspaces.

  • Complete documentation of all functions and methods.

  • Complete doctests of all functions and methods.

Future ideas (hard):

  • Support for general triangle groups
  • Support for "congruence" subgroups