# 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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Conjectural Slopes of Hecke Polynomial