The dimension formulas and implementations in this module grew out of a program that Bruce Caskel wrote (around 1996) in PARI, which Kevin Buzzard subsequently extended. I (William Stein) then implemented it in C++ for HECKE. I also implemented it in MAGMA. Also, the functions for dimensions of spaces with nontrivial character are based on a paper of Cohen and Oesterle (Springer Lecture notes in math, volume 627, pages 69-78). I asked Cohen about proofs of the formulas for nontrivial character, and learned that they have never been published.
The module modular.dims defines the following functions:
group, [k=2]) |
The dimension of the space of cusp forms for the congruence subgroup group.
eps, [k=2]) |
The dimension of the space of cusp forms of weight k and character eps.
INPUT: eps -- a Dirichlet character k -- int, a weight >= 2. OUTPUT: int -- the dimension
sage: from sage.modular.dirichlet import * sage: G = DirichletGroup(13) sage: e = G.gen() sage: e.order() 12 sage: dimension_cusp_forms_eps(e,2) 0 sage: dimension_cusp_forms_eps(e**2,2) 1
group, [k=2]) |
The dimension of the space of eisenstein series for the given congruence subgroup.
eps, [k=None], [p=2]) |
Dimension of the new subspace (or p-new subspace) of cusp forms of weight k and character eps.
N, [k=0], [p=2]) |
Dimension of the p-new subspace of S_k(Gamma_0(N)). If p=0, dimension of the new subspace.
N, [k=0], [p=2]) |
group, [k=2]) |
The dimension of the new space of cusp forms for the congruence subgroup group.
N) |
The index
.
N) |
The index
.
See About this document... for information on suggesting changes.