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jacobi_space.py
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jacobi_space.py
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r"""
Modular forms for Hecke triangle groups
AUTHORS:
- Jonas Jermann (2013): initial version
"""
#*****************************************************************************
# Copyright (C) 2013-2014 Jonas Jermann <jjermann2@gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.rings.all import ZZ, QQ, infinity
from sage.modules.module import Module
from sage.categories.all import Modules
from sage.modules.free_module import FreeModule
from sage.modules.free_module_element import vector
from sage.structure.unique_representation import UniqueRepresentation
from sage.misc.cachefunc import cached_method
from hecke_triangle_groups import HeckeTriangleGroup
from abstract_space import FormsSpace_abstract
def canonical_parameters(group, base_ring, k, ep, n=None):
r"""
Return a canonical version of the parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import canonical_parameters
sage: canonical_parameters(5, ZZ, 20/3, int(1))
(Hecke triangle group for n = 5, Integer Ring, 20/3, 1, 5)
sage: canonical_parameters(infinity, ZZ, 2, int(-1))
(Hecke triangle group for n = +Infinity, Integer Ring, 2, -1, +Infinity)
"""
if not (n is None):
group = n
if (group == infinity):
group = HeckeTriangleGroup(infinity)
else:
try:
group = HeckeTriangleGroup(ZZ(group))
except TypeError:
group = HeckeTriangleGroup(group.n())
n = group.n()
k = QQ(k)
if (ep == None):
if (n == infinity):
ep = (-1)**(k/ZZ(2))
elif (ZZ(2).divides(n)):
ep = (-1)**(k*ZZ(n-2)/ZZ(4))
else:
ep = (-1)**(k*ZZ(n-2)/ZZ(2))
ep = ZZ(ep)
if (n == infinity):
num = (k-(1-ep)) / ZZ(4)
else:
num = (k-(1-ep)*n/(n-2)) * (n-2) / ZZ(4)
try:
num = ZZ(num)
except TypeError:
pass
#raise ValueError("Invalid or non-occuring weight k={}, ep={}!".format(k,ep))
return (group, base_ring, k, ep, n)
class QuasiMeromorphicJacobiForms(FormsSpace_abstract, Module, UniqueRepresentation):
r"""
Module of (Hecke) quasi meromorphic modular forms
for the given group, base ring, weight and multiplier
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, k=QQ(0), ep=None, n=None):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import (canonical_parameters, QuasiMeromorphicModularForms)
sage: (group, base_ring, k, ep, n) = canonical_parameters(5, ZZ, 20/3, int(1))
sage: QuasiMeromorphicModularForms(5, ZZ, 20/3, int(1)) == QuasiMeromorphicModularForms(group, base_ring, k, ep, n)
True
"""
(group, base_ring, k, ep, n) = canonical_parameters(group, base_ring, k, ep, n)
return super(FormsSpace_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
def __init__(self, group, base_ring, k, ep, n):
r"""
Return the Module of (Hecke) quasi meromorphic modular forms
of weight ``k`` with multiplier ``ep`` for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: MF = QuasiMeromorphicModularForms(5, ZZ, 20/3, 1)
sage: MF
QuasiMeromorphicModularForms(n=5, k=20/3, ep=1) over Integer Ring
sage: MF.analytic_type()
quasi meromorphic modular
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.ambient_space() == MF
True
"""
FormsSpace_abstract.__init__(self, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["jacobi", "quasi", "mero"])
class QuasiWeakJacobiForms(FormsSpace_abstract, Module, UniqueRepresentation):
r"""
Module of (Hecke) quasi weakly holomorphic modular forms
for the given group, base ring, weight and multiplier
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, k=QQ(0), ep=None, n=None):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import (canonical_parameters, QuasiWeakModularForms)
sage: (group, base_ring, k, ep, n) = canonical_parameters(4, ZZ, 8, -1)
sage: QuasiWeakModularForms(4, ZZ, 8, -1) == QuasiWeakModularForms(group, base_ring, k, ep, n)
True
"""
(group, base_ring, k, ep, n) = canonical_parameters(group, base_ring, k, ep, n)
return super(FormsSpace_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
def __init__(self, group, base_ring, k, ep, n):
r"""
Return the Module of (Hecke) quasi weakly holomorphic modular forms
of weight ``k`` with multiplier ``ep`` for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import QuasiWeakModularForms
sage: MF = QuasiWeakModularForms(4, ZZ, 8, 1)
sage: MF
QuasiWeakModularForms(n=4, k=8, ep=1) over Integer Ring
sage: MF.analytic_type()
quasi weakly holomorphic modular
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.is_ambient()
True
"""
FormsSpace_abstract.__init__(self, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["jacobi", "quasi", "weak"])
class QuasiJacobiForms(FormsSpace_abstract, Module, UniqueRepresentation):
r"""
Module of (Hecke) quasi modular forms
for the given group, base ring, weight and multiplier
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, k=QQ(0), ep=None, n=None):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import (canonical_parameters, QuasiModularForms)
sage: (group, base_ring, k, ep, n) = canonical_parameters(5, ZZ, 10/3, -1)
sage: QuasiModularForms(5, ZZ, 10/3) == QuasiModularForms(group, base_ring, k, ep, n)
True
"""
(group, base_ring, k, ep, n) = canonical_parameters(group, base_ring, k, ep, n)
return super(FormsSpace_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
def __init__(self, group, base_ring, k, ep, n):
r"""
Return the Module of (Hecke) quasi modular forms
of weight ``k`` with multiplier ``ep`` for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: MF = QuasiModularForms(5, ZZ, 20/3, 1)
sage: MF
QuasiModularForms(n=5, k=20/3, ep=1) over Integer Ring
sage: MF.analytic_type()
quasi modular
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.is_ambient()
True
"""
FormsSpace_abstract.__init__(self, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["jacobi", "quasi", "holo"])
class QuasiCuspJacobiForms(FormsSpace_abstract, Module, UniqueRepresentation):
r"""
Module of (Hecke) quasi cusp forms
for the given group, base ring, weight and multiplier
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, k=QQ(0), ep=None, n=None):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import (canonical_parameters, QuasiCuspForms)
sage: (group, base_ring, k, ep, n) = canonical_parameters(8, ZZ, 16/3, None)
sage: QuasiCuspForms(8, ZZ, 16/3) == QuasiCuspForms(group, base_ring, k, ep, n)
True
"""
(group, base_ring, k, ep, n) = canonical_parameters(group, base_ring, k, ep, n)
return super(FormsSpace_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
def __init__(self, group, base_ring, k, ep, n):
r"""
Return the Module of (Hecke) quasi cusp forms
of weight ``k`` with multiplier ``ep`` for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import QuasiCuspForms
sage: MF = QuasiCuspForms(8, ZZ, 16/3)
sage: MF
QuasiCuspForms(n=8, k=16/3, ep=1) over Integer Ring
sage: MF.analytic_type()
quasi cuspidal
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.is_ambient()
True
sage: QuasiCuspForms(n=infinity)
QuasiCuspForms(n=+Infinity, k=0, ep=1) over Integer Ring
"""
FormsSpace_abstract.__init__(self, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["jacobi", "quasi", "cusp"])
class MeromorphicJacobiForms(FormsSpace_abstract, Module, UniqueRepresentation):
r"""
Module of (Hecke) meromorphic modular forms
for the given group, base ring, weight and multiplier
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, k=QQ(0), ep=None, n=None):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import (canonical_parameters, MeromorphicModularForms)
sage: (group, base_ring, k, ep, n) = canonical_parameters(3, ZZ, 0, 1)
sage: MeromorphicModularForms() == MeromorphicModularForms(group, base_ring, k, ep, n)
True
"""
(group, base_ring, k, ep, n) = canonical_parameters(group, base_ring, k, ep, n)
return super(FormsSpace_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
def __init__(self, group, base_ring, k, ep, n):
r"""
Return the Module of (Hecke) meromorphic modular forms
of weight ``k`` with multiplier ``ep`` for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import MeromorphicModularForms
sage: MF = MeromorphicModularForms()
sage: MF
MeromorphicModularForms(n=3, k=0, ep=1) over Integer Ring
sage: MF.analytic_type()
meromorphic modular
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.is_ambient()
True
"""
FormsSpace_abstract.__init__(self, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["jacobi", "mero"])
class WeakJacobiForms(FormsSpace_abstract, Module, UniqueRepresentation):
r"""
Module of (Hecke) weakly holomorphic modular forms
for the given group, base ring, weight and multiplier
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, k=QQ(0), ep=None, n=None):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import (canonical_parameters, WeakModularForms)
sage: (group, base_ring, k, ep, n) = canonical_parameters(5, CC, 20/3, None)
sage: WeakModularForms(5, CC, 20/3) == WeakModularForms(group, base_ring, k, ep, n)
True
"""
(group, base_ring, k, ep, n) = canonical_parameters(group, base_ring, k, ep, n)
return super(FormsSpace_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
def __init__(self, group, base_ring, k, ep, n):
r"""
Return the Module of (Hecke) weakly holomorphic modular forms
of weight ``k`` with multiplier ``ep`` for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import WeakModularForms
sage: MF = WeakModularForms(5, CC, 20/3)
sage: MF
WeakModularForms(n=5, k=20/3, ep=1) over Complex Field with 53 bits of precision
sage: MF.analytic_type()
weakly holomorphic modular
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Complex Field with 53 bits of precision
"""
FormsSpace_abstract.__init__(self, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["jacobi", "weak"])
class JacobiForms(FormsSpace_abstract, Module, UniqueRepresentation):
r"""
Module of (Hecke) modular forms
for the given group, base ring, weight and multiplier
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, k=QQ(0), ep=None, n=None):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import (canonical_parameters, ModularForms)
sage: (group, base_ring, k, ep, n) = canonical_parameters(3, ZZ, 0, None)
sage: ModularForms() == ModularForms(group, base_ring, k, ep, n)
True
"""
(group, base_ring, k, ep, n) = canonical_parameters(group, base_ring, k, ep, n)
return super(FormsSpace_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
def __init__(self, group, base_ring, k, ep, n):
r"""
Return the Module of (Hecke) modular forms
of weight ``k`` with multiplier ``ep`` for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms()
sage: MF
ModularForms(n=3, k=0, ep=1) over Integer Ring
sage: MF.analytic_type()
modular
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.module()
Vector space of dimension 1 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.ambient_module() == MF.module()
True
sage: MF.is_ambient()
True
sage: MF = ModularForms(n=infinity, k=8)
sage: MF
ModularForms(n=+Infinity, k=8, ep=1) over Integer Ring
sage: MF.analytic_type()
modular
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
"""
FormsSpace_abstract.__init__(self, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type = self.AT(["jacobi", "holo"])
class CuspJacobiForms(FormsSpace_abstract, Module, UniqueRepresentation):
r"""
Module of (Hecke) cusp forms
for the given group, base ring, weight and multiplier
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, k=QQ(0), ep=None, n=None):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import (canonical_parameters, CuspForms)
sage: (group, base_ring, k, ep, n) = canonical_parameters(6, ZZ, 6, 1)
sage: CuspForms(6, ZZ, 6, 1) == CuspForms(group, base_ring, k, ep, n)
True
"""
(group, base_ring, k, ep, n) = canonical_parameters(group, base_ring, k, ep, n)
return super(FormsSpace_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
def __init__(self, group, base_ring, k, ep, n):
r"""
Return the Module of (Hecke) cusp forms
of weight ``k`` with multiplier ``ep`` for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import CuspForms
sage: MF = CuspForms(6, ZZ, 6, 1)
sage: MF
CuspForms(n=6, k=6, ep=1) over Integer Ring
sage: MF.analytic_type()
cuspidal
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.module()
Vector space of dimension 1 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.ambient_module() == MF.module()
True
sage: MF.is_ambient()
True
"""
FormsSpace_abstract.__init__(self, group=group, base_ring=base_ring, k=k, ep=ep, n=n)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["jacobi", "cusp"])