Dimensions of spaces of modular forms

AUTHORS:

  • William Stein
  • Jordi Quer

ACKNOWLEDGEMENT: The dimension formulas and implementations in this module grew out of a program that Bruce Kaskel 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 (that has no proofs) by Cohen and Oesterle (Springer Lecture notes in math, volume 627, pages 69-78). The formulas for \(\Gamma_H(N)\) were found and implemented by Jordi Quer.

The formulas here are more complete than in Hecke or Magma.

Currently the input to each function below is an integer and either a Dirichlet character \(\varepsilon\) or a finite index subgroup of \({\rm SL}_2(\ZZ)\). If the input is a Dirichlet character \(\varepsilon\), the dimensions are for subspaces of \(M_k(\Gamma_1(N), \varepsilon)\), where \(N\) is the modulus of \(\varepsilon\).

These functions mostly call the methods dimension_cusp_forms, dimension_modular_forms and so on of the corresponding congruence subgroup classes.

sage.modular.dims.CO_delta(r, p, N, eps)

This is used as an intermediate value in computations related to the paper of Cohen-Oesterle.

INPUT:

  • r - positive integer
  • p - a prime
  • N - positive integer
  • eps - character

OUTPUT: element of the base ring of the character

EXAMPLES:

sage: G.<eps> = DirichletGroup(7)
sage: sage.modular.dims.CO_delta(1,5,7,eps^3)
2
sage.modular.dims.CO_nu(r, p, N, eps)

This is used as an intermediate value in computations related to the paper of Cohen-Oesterle.

INPUT:

  • r - positive integer
  • p - a prime
  • N - positive integer
  • eps - character

OUTPUT: element of the base ring of the character

EXAMPLES:

sage: G.<eps> = DirichletGroup(7)
sage: G.<eps> = DirichletGroup(7)
sage: sage.modular.dims.CO_nu(1,7,7,eps)
-1
sage.modular.dims.CohenOesterle(eps, k)

Compute the Cohen-Oesterle function associate to eps, \(k\). This is a summand in the formula for the dimension of the space of cusp forms of weight \(2\) with character \(\varepsilon\).

INPUT:

  • eps - Dirichlet character
  • k - integer

OUTPUT: element of the base ring of eps.

EXAMPLES:

sage: G.<eps> = DirichletGroup(7)
sage: sage.modular.dims.CohenOesterle(eps, 2)
-2/3
sage: sage.modular.dims.CohenOesterle(eps, 4)
-1
sage.modular.dims.dimension_cusp_forms(X, k=2)

The dimension of the space of cusp forms for the given congruence subgroup or Dirichlet character.

INPUT:

  • X - congruence subgroup or Dirichlet character or integer
  • k - weight (integer)

EXAMPLES:

sage: dimension_cusp_forms(5,4)
1
sage: dimension_cusp_forms(Gamma0(11),2)
1
sage: dimension_cusp_forms(Gamma1(13),2)
2
sage: dimension_cusp_forms(DirichletGroup(13).0^2,2)
1
sage: dimension_cusp_forms(DirichletGroup(13).0,3)
1
sage: dimension_cusp_forms(Gamma0(11),2)
1
sage: dimension_cusp_forms(Gamma0(11),0)
0
sage: dimension_cusp_forms(Gamma0(1),12)
1
sage: dimension_cusp_forms(Gamma0(1),2)
0
sage: dimension_cusp_forms(Gamma0(1),4)
0
sage: dimension_cusp_forms(Gamma0(389),2)
32
sage: dimension_cusp_forms(Gamma0(389),4)
97
sage: dimension_cusp_forms(Gamma0(2005),2)
199
sage: dimension_cusp_forms(Gamma0(11),1)
0
sage: dimension_cusp_forms(Gamma1(11),2)
1
sage: dimension_cusp_forms(Gamma1(1),12)
1
sage: dimension_cusp_forms(Gamma1(1),2)
0
sage: dimension_cusp_forms(Gamma1(1),4)
0
sage: dimension_cusp_forms(Gamma1(389),2)
6112
sage: dimension_cusp_forms(Gamma1(389),4)
18721
sage: dimension_cusp_forms(Gamma1(2005),2)
159201
sage: dimension_cusp_forms(Gamma1(11),1)
0
sage: e = DirichletGroup(13).0
sage: e.order()
12
sage: dimension_cusp_forms(e,2)
0
sage: dimension_cusp_forms(e^2,2)
1

Check that trac ticket #12640 is fixed:

sage: dimension_cusp_forms(DirichletGroup(1)(1), 12)
1
sage: dimension_cusp_forms(DirichletGroup(2)(1), 24)
5
sage.modular.dims.dimension_eis(X, k=2)

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

INPUT:

  • X - congruence subgroup or Dirichlet character or integer
  • k - weight (integer)

EXAMPLES:

sage: dimension_eis(5,4)
2
sage: dimension_eis(Gamma0(11),2)
1
sage: dimension_eis(Gamma1(13),2)
11
sage: dimension_eis(Gamma1(2006),2)
3711
sage: e = DirichletGroup(13).0
sage: e.order()
12
sage: dimension_eis(e,2)
0
sage: dimension_eis(e^2,2)
2
sage: e = DirichletGroup(13).0
sage: e.order()
12
sage: dimension_eis(e,2)
0
sage: dimension_eis(e^2,2)
2
sage: dimension_eis(e,13)
2
sage: G = DirichletGroup(20)
sage: dimension_eis(G.0,3)
4
sage: dimension_eis(G.1,3)
6
sage: dimension_eis(G.1^2,2)
6
sage: G = DirichletGroup(200)
sage: e = prod(G.gens(), G(1))
sage: e.conductor()
200
sage: dimension_eis(e,2)
4
sage: dimension_modular_forms(Gamma1(4), 11)
6
sage.modular.dims.dimension_modular_forms(X, k=2)

The dimension of the space of cusp forms for the given congruence subgroup (either \(\Gamma_0(N)\), \(\Gamma_1(N)\), or \(\Gamma_H(N)\)) or Dirichlet character.

INPUT:

  • X - congruence subgroup or Dirichlet character
  • k - weight (integer)

EXAMPLES:

sage: dimension_modular_forms(Gamma0(11),2)
2
sage: dimension_modular_forms(Gamma0(11),0)
1
sage: dimension_modular_forms(Gamma1(13),2)
13
sage: dimension_modular_forms(GammaH(11, [10]), 2)
10
sage: dimension_modular_forms(GammaH(11, [10]))
10
sage: dimension_modular_forms(GammaH(11, [10]), 4)
20
sage: e = DirichletGroup(20).1
sage: dimension_modular_forms(e,3)
9
sage: dimension_cusp_forms(e,3)
3
sage: dimension_eis(e,3)
6
sage: dimension_modular_forms(11,2)
2
sage.modular.dims.dimension_new_cusp_forms(X, k=2, p=0)

Return the dimension of the new (or \(p\)-new) subspace of cusp forms for the character or group \(X\).

INPUT:

  • X - integer, congruence subgroup or Dirichlet character
  • k - weight (integer)
  • p - 0 or a prime

EXAMPLES:

sage: dimension_new_cusp_forms(100,2)
1
sage: dimension_new_cusp_forms(Gamma0(100),2)
1
sage: dimension_new_cusp_forms(Gamma0(100),4)
5
sage: dimension_new_cusp_forms(Gamma1(100),2)
141
sage: dimension_new_cusp_forms(Gamma1(100),4)
463
sage: dimension_new_cusp_forms(DirichletGroup(100).1^2,2)
2
sage: dimension_new_cusp_forms(DirichletGroup(100).1^2,4)
8
sage: sum(dimension_new_cusp_forms(e,3) for e in DirichletGroup(30))
12
sage: dimension_new_cusp_forms(Gamma1(30),3)
12

Check that trac ticket #12640 is fixed:

sage: dimension_new_cusp_forms(DirichletGroup(1)(1), 12)
1
sage: dimension_new_cusp_forms(DirichletGroup(2)(1), 24)
1
sage.modular.dims.eisen(p)

Return the Eisenstein number \(n\) which is the numerator of \((p-1)/12\).

INPUT:

  • p - a prime

OUTPUT: Integer

EXAMPLES:

sage: [(p,sage.modular.dims.eisen(p)) for p in prime_range(24)]
[(2, 1), (3, 1), (5, 1), (7, 1), (11, 5), (13, 1), (17, 4), (19, 3), (23, 11)]
sage.modular.dims.sturm_bound(level, weight=2)

Returns the Sturm bound for modular forms with given level and weight. For more details, see the documentation for the sturm_bound method of sage.modular.arithgroup.CongruenceSubgroup objects.

INPUT:

  • level - an integer (interpreted as a level for Gamma0) or a congruence subgroup
  • weight - an integer \(\geq 2\) (default: 2)

EXAMPLES:

sage: sturm_bound(11,2)
2
sage: sturm_bound(389,2)
65
sage: sturm_bound(1,12)
1
sage: sturm_bound(100,2)
30
sage: sturm_bound(1,36)
3
sage: sturm_bound(11)
2