def sturm_bound(level, weight=2):
r"""
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
"""
if is_ArithmeticSubgroup(level):
if level.is_congruence():
return level.sturm_bound(weight)
else:
raise ValueError("No Sturm bound defined for noncongruence subgroups")
if isinstance(level, (int, long, Integer)):
return Gamma0(level).sturm_bound(weight)
def sturm_bound(level, weight=2):
r"""
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
"""
if is_ArithmeticSubgroup(level):
if level.is_congruence():
return level.sturm_bound(weight)
else:
raise ValueError, "No Sturm bound defined for noncongruence subgroups"
if isinstance(level, (int, long, Integer)):
return Gamma0(level).sturm_bound(weight)
def dimension_modular_forms(X, k=2):
r"""
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
"""
if isinstance(X, integer_types + (Integer, )):
return Gamma0(X).dimension_modular_forms(k)
elif is_ArithmeticSubgroup(X):
return X.dimension_modular_forms(k)
elif isinstance(X, dirichlet.DirichletCharacter):
return Gamma1(X.modulus()).dimension_modular_forms(k, eps=X)
else:
raise TypeError(
"Argument 1 must be an integer, a Dirichlet character or an arithmetic subgroup."
)
def dimension_modular_forms(X, k=2):
r"""
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
"""
if isinstance(X, (int, long, Integer)):
return Gamma0(X).dimension_modular_forms(k)
elif is_ArithmeticSubgroup(X):
return X.dimension_modular_forms(k)
elif isinstance(X,dirichlet.DirichletCharacter):
return Gamma1(X.modulus()).dimension_modular_forms(k, eps=X)
else:
raise TypeError("Argument 1 must be an integer, a Dirichlet character or an arithmetic subgroup.")
def 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
"""
if is_ArithmeticSubgroup(X):
return X.dimension_eis(k)
elif isinstance(X, dirichlet.DirichletCharacter):
return Gamma1(X.modulus()).dimension_eis(k, X)
elif isinstance(X, (int, long, Integer)):
return Gamma0(X).dimension_eis(k)
else:
raise TypeError("Argument in dimension_eis must be an integer, a Dirichlet character, or a finite index subgroup of SL2Z (got %s)" % X)
def dimension_cusp_forms(X, k=2):
r"""
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
"""
if isinstance(X, dirichlet.DirichletCharacter):
N = X.modulus()
if N <= 2:
return Gamma0(N).dimension_cusp_forms(k)
else:
return Gamma1(N).dimension_cusp_forms(k, X)
elif is_ArithmeticSubgroup(X):
return X.dimension_cusp_forms(k)
elif isinstance(X, (Integer,int,long)):
return Gamma0(X).dimension_cusp_forms(k)
else:
raise TypeError("Argument 1 must be a Dirichlet character, an integer or a finite index subgroup of SL2Z")
def 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
"""
if is_ArithmeticSubgroup(X):
return X.dimension_eis(k)
elif isinstance(X, dirichlet.DirichletCharacter):
return Gamma1(X.modulus()).dimension_eis(k, X)
elif isinstance(X, (int, long, Integer)):
return Gamma0(X).dimension_eis(k)
else:
raise TypeError, "Argument in dimension_eis must be an integer, a Dirichlet character, or a finite index subgroup of SL2Z (got %s)" % X
def dimension_cusp_forms(X, k=2):
r"""
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
"""
if isinstance(X, dirichlet.DirichletCharacter):
N = X.modulus()
if N <= 2:
return Gamma0(N).dimension_cusp_forms(k)
else:
return Gamma1(N).dimension_cusp_forms(k, X)
elif is_ArithmeticSubgroup(X):
return X.dimension_cusp_forms(k)
elif isinstance(X, (Integer, int, long)):
return Gamma0(X).dimension_cusp_forms(k)
else:
raise TypeError, "Argument 1 must be a Dirichlet character, an integer or a finite index subgroup of SL2Z"
