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# 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 #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 #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

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