7.4.1 modular.dims - Dimensions of spaces of modular forms

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:

dimension_cusp_forms( group, [k=2])

The dimension of the space of cusp forms for the congruence subgroup group.

dimension_cusp_forms_eps( 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

dimension_eis( group, [k=2])

The dimension of the space of eisenstein series for the given congruence subgroup.

dimension_new_cusp_forms( eps, [k=None], [p=2])

Dimension of the new subspace (or p-new subspace) of cusp forms of weight k and character eps.

dimension_new_cusp_forms_gamma0( 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.

dimension_new_cusp_forms_gamma1( N, [k=0], [p=2])
Dimension of the p-new subspace of S_k(Gamma_1(N)). If p=0, dimension of the new subspace.

dimension_new_cusp_forms_group( group, [k=2])

The dimension of the new space of cusp forms for the congruence subgroup group.

idxG0( N)

The index $ [\Gamma_0(N):\SL _2(\mathbf{Z})]$ .

idxG1( N)

The index $ [\Gamma_1(N):\SL _2(\mathbf{Z})]$ .

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