def eisenstein_params(self):
"""
Return parameters that define all Eisenstein series in self.
OUTPUT: an immutable Sequence
EXAMPLES::
sage: m = ModularForms(Gamma0(22), 2)
sage: v = m.eisenstein_params(); v
[(Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 2), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 11), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 22)]
sage: type(v)
<class 'sage.structure.sequence.Sequence_generic'>
"""
try:
return self.__eisenstein_params
except AttributeError:
eps = self.character()
if eps is None:
if arithgroup.is_Gamma1(self.group()):
eps = self.level()
else:
raise NotImplementedError
params = eis_series.compute_eisenstein_params(eps, self.weight())
self.__eisenstein_params = Sequence(params, immutable=True)
return self.__eisenstein_params
def eisenstein_params(self):
"""
Return parameters that define all Eisenstein series in self.
OUTPUT: an immutable Sequence
EXAMPLES::
sage: m = ModularForms(Gamma0(22), 2)
sage: v = m.eisenstein_params(); v
[(Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 2), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 11), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 22)]
sage: type(v)
<class 'sage.structure.sequence.Sequence_generic'>
"""
try:
return self.__eisenstein_params
except AttributeError:
eps = self.character()
if eps == None:
if arithgroup.is_Gamma1(self.group()):
eps = self.level()
else:
raise NotImplementedError
params = eis_series.compute_eisenstein_params(eps, self.weight())
self.__eisenstein_params = Sequence(params, immutable=True)
return self.__eisenstein_params
def parameters(self):
r"""
Return a list of parameters for each Eisenstein series
spanning self. That is, for each such series, return a triple
of the form (`\psi`, `\chi`, level), where `\psi` and `\chi`
are the characters defining the Eisenstein series, and level
is the smallest level at which this series occurs.
EXAMPLES::
sage: ModularForms(24,2).eisenstein_submodule().parameters()
[(Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, 2),
...
Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, 24)]
sage: EisensteinForms(12,6).parameters()[-1]
(Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1, Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1, 12)
sage: pars = ModularForms(DirichletGroup(24).0,3).eisenstein_submodule().parameters()
sage: [(x[0].values_on_gens(),x[1].values_on_gens(),x[2]) for x in pars]
[((1, 1, 1), (-1, 1, 1), 1),
((1, 1, 1), (-1, 1, 1), 2),
((1, 1, 1), (-1, 1, 1), 3),
((1, 1, 1), (-1, 1, 1), 6),
((-1, 1, 1), (1, 1, 1), 1),
((-1, 1, 1), (1, 1, 1), 2),
((-1, 1, 1), (1, 1, 1), 3),
((-1, 1, 1), (1, 1, 1), 6)]
sage: EisensteinForms(DirichletGroup(24).0,1).parameters()
[(Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 1), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 2), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 3), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 6)]
"""
try:
return self.__parameters
except AttributeError:
char = self._parameters_character()
if char is None:
P = eis_series.compute_eisenstein_params(
self.level(), self.weight())
else:
P = eis_series.compute_eisenstein_params(char, self.weight())
self.__parameters = P
return P
def parameters(self):
r"""
Return a list of parameters for each Eisenstein series
spanning self. That is, for each such series, return a triple
of the form (`\psi`, `\chi`, level), where `\psi` and `\chi`
are the characters defining the Eisenstein series, and level
is the smallest level at which this series occurs.
EXAMPLES::
sage: ModularForms(24,2).eisenstein_submodule().parameters()
[(Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, 2),
...
Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, 24)]
sage: EisensteinForms(12,6).parameters()[-1]
(Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1, Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1, 12)
sage: pars = ModularForms(DirichletGroup(24).0,3).eisenstein_submodule().parameters()
sage: [(x[0].values_on_gens(),x[1].values_on_gens(),x[2]) for x in pars]
[((1, 1, 1), (-1, 1, 1), 1),
((1, 1, 1), (-1, 1, 1), 2),
((1, 1, 1), (-1, 1, 1), 3),
((1, 1, 1), (-1, 1, 1), 6),
((-1, 1, 1), (1, 1, 1), 1),
((-1, 1, 1), (1, 1, 1), 2),
((-1, 1, 1), (1, 1, 1), 3),
((-1, 1, 1), (1, 1, 1), 6)]
sage: EisensteinForms(DirichletGroup(24).0,1).parameters()
[(Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 1), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 2), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 3), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 6)]
"""
try:
return self.__parameters
except AttributeError:
char = self._parameters_character()
if char is None:
P = eis_series.compute_eisenstein_params(self.level(), self.weight())
else:
P = eis_series.compute_eisenstein_params(char, self.weight())
self.__parameters = P
return P
