def __init__(self, group, base_ring, k, ep):
r"""
Return the zero Module for the zero form of weight ``k`` with multiplier ``ep``
for the given ``group`` and ``base_ring``.
The ZeroForm space coerces into any forms space or ring with a compatible group.
EXAMPLES::
sage: MF = ZeroForm(6, CC, 3, -1)
sage: MF
ZeroForms(n=6, k=3, ep=-1) over Complex Field with 53 bits of precision
sage: MF.analytic_type()
zero
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Complex Field with 53 bits of precision
sage: MF.module()
Vector space of dimension 0 over Fraction Field of Univariate Polynomial Ring in d over Complex Field with 53 bits of precision
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)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT([])
self._module = FreeModule(self.coeff_ring(), self.dimension())
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"])
def __init__(self, group, base_ring, k, ep):
r"""
Return the Module of (Hecke) cusp forms
of weight ``k`` with multiplier ``ep`` for the given ``group`` and ``base_ring``.
EXAMPLES::
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)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["cusp"])
self._module = FreeModule(self.coeff_ring(), self.dimension())
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"])
def __init__(self, group, base_ring, k, ep, n):
r"""
Return the zero Module for the zero form of weight ``k`` with multiplier ``ep``
for the given ``group`` and ``base_ring``.
The ZeroForm space coerces into any forms space or ring with a compatible group.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import ZeroForm
sage: MF = ZeroForm(6, CC, 3, -1)
sage: MF
ZeroForms(n=6, k=3, ep=-1) over Complex Field with 53 bits of precision
sage: MF.analytic_type()
zero
sage: MF.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Complex Field with 53 bits of precision
sage: MF.module()
Vector space of dimension 0 over Fraction Field of Univariate Polynomial Ring in d over Complex Field with 53 bits of precision
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([])
self._module = FreeModule(self.coeff_ring(), self.dimension())
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(["cusp"])
self._module = FreeModule(self.coeff_ring(), self.dimension())
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(["quasi", "cusp"])
self._module = FreeModule(self.coeff_ring(), self.dimension())
def __init__(self, group, base_ring, k, ep):
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: 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)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["weak"])
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(["weak"])
def __init__(self, group, base_ring, k, ep):
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: MF = QMModularForms(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)
Module.__init__(self, base=self.coeff_ring())
self._analytic_type=self.AT(["quasi", "mero"])
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"])
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(["mero"])
def __init__(self, ambient_space, basis, check):
r"""
Return the Submodule of (Hecke) forms in ``ambient_space`` for the given ``basis``.
INPUT:
- ``ambient_space`` -- An ambient forms space.
- ``basis`` -- A tuple of (not necessarily linearly independent)
elements of ``ambient_space``.
- ``check`` -- If ``True`` (default) then a maximal linearly
independent subset of ``basis`` is choosen. Otherwise
it is assumed that ``basis`` is linearly independent.
OUTPUT:
The corresponding submodule.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import ModularForms, QuasiCuspForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: MF
ModularForms(n=6, k=20, ep=1) over Integer Ring
sage: MF.dimension()
4
sage: subspace = MF.subspace([MF.Delta()*MF.E4()^2, MF.gen(0), 2*MF.gen(0)])
sage: subspace
Subspace of dimension 2 of ModularForms(n=6, k=20, ep=1) over Integer Ring
sage: subspace.analytic_type()
modular
sage: subspace.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.module()
Vector space of degree 4 and dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
Basis matrix:
[ 1 0 0 0]
[ 0 1 13/(18*d) 103/(432*d^2)]
sage: subspace.ambient_module()
Vector space of dimension 4 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.ambient_module() == MF.module()
True
sage: subspace.ambient_space() == MF
True
sage: subspace.basis()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
sage: subspace.basis()[0].parent() == MF
True
sage: subspace.gens()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
sage: subspace.gens()[0].parent() == subspace
True
sage: subspace.is_ambient()
False
sage: MF = QuasiCuspForms(n=infinity, k=12, ep=1)
sage: MF.dimension()
4
sage: subspace = MF.subspace([MF.Delta(), MF.E4()*MF.f_inf()*MF.E2()*MF.f_i(), MF.E4()*MF.f_inf()*MF.E2()^2, MF.E4()*MF.f_inf()*(MF.E4()-MF.E2()^2)])
sage: subspace.default_prec(3)
sage: subspace
Subspace of dimension 3 of QuasiCuspForms(n=+Infinity, k=12, ep=1) over Integer Ring
sage: subspace.gens()
[q + 24*q^2 + O(q^3), q - 24*q^2 + O(q^3), q - 8*q^2 + O(q^3)]
"""
FormsSpace_abstract.__init__(self,
group=ambient_space.group(),
base_ring=ambient_space.base_ring(),
k=ambient_space.weight(),
ep=ambient_space.ep(),
n=ambient_space.hecke_n())
Module.__init__(self, base=self.coeff_ring())
self._ambient_space = ambient_space
self._basis = [v for v in basis]
# self(v) instead would somehow mess up the coercion model
self._gens = [self._element_constructor_(v) for v in basis]
self._module = ambient_space._module.submodule(
[ambient_space.coordinate_vector(v) for v in basis])
# TODO: get the analytic type from the basis
#self._analytic_type=self.AT(["quasi", "mero"])
self._analytic_type = ambient_space._analytic_type
def __init__(self, ambient_space, basis):
r"""
Return the Submodule of (Hecke) forms in ``ambient_space`` for the given ``basis``.
INPUT:
- ``ambient_space`` - An ambient forms space.
- ``basis``` - A tuple of linearly independent elements of ``ambient_space``.
OUTPUT:
The corresponding submodule.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=20, ep=1)
sage: MF
ModularForms(n=6, k=20, ep=1) over Integer Ring
sage: MF.dimension()
4
sage: subspace = MF.subspace([MF.Delta()*MF.E4()^2, MF.gen(0)])
sage: subspace
Subspace of dimension 2 of ModularForms(n=6, k=20, ep=1) over Integer Ring
sage: subspace.analytic_type()
modular
sage: subspace.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.module()
Vector space of degree 4 and dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
Basis matrix:
[ 1 0 0 0]
[ 0 1 13/(18*d) 103/(432*d^2)]
sage: subspace.ambient_module()
Vector space of dimension 4 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.ambient_module() == MF.module()
True
sage: subspace.ambient_space() == MF
True
sage: subspace.basis()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
sage: subspace.basis()[0].parent() == MF
True
sage: subspace.gens()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
sage: subspace.gens()[0].parent() == subspace
True
sage: subspace.is_ambient()
False
"""
FormsSpace_abstract.__init__(self, group=ambient_space.group(), base_ring=ambient_space.base_ring(), k=ambient_space.weight(), ep=ambient_space.ep())
Module.__init__(self, base=self.coeff_ring())
self._ambient_space = ambient_space
self._basis = [v for v in basis]
# self(v) instead would somehow mess up the coercion model
self._gens = [self._element_constructor_(v) for v in basis]
self._module = ambient_space._module.submodule([ambient_space.coordinate_vector(v) for v in basis])
# TODO: get the analytic type from the basis
#self._analytic_type=self.AT(["quasi", "mero"])
self._analytic_type = ambient_space._analytic_type
def __init__(self, ambient_space, basis, check):
r"""
Return the Submodule of (Hecke) forms in ``ambient_space`` for the given ``basis``.
INPUT:
- ``ambient_space`` -- An ambient forms space.
- ``basis`` -- A tuple of (not necessarily linearly independent)
elements of ``ambient_space``.
- ``check`` -- If ``True`` (default) then a maximal linearly
independent subset of ``basis`` is choosen. Otherwise
it is assumed that ``basis`` is linearly independent.
OUTPUT:
The corresponding submodule.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.space import ModularForms, QuasiCuspForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: MF
ModularForms(n=6, k=20, ep=1) over Integer Ring
sage: MF.dimension()
4
sage: subspace = MF.subspace([MF.Delta()*MF.E4()^2, MF.gen(0), 2*MF.gen(0)])
sage: subspace
Subspace of dimension 2 of ModularForms(n=6, k=20, ep=1) over Integer Ring
sage: subspace.analytic_type()
modular
sage: subspace.category()
Category of vector spaces over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.module()
Vector space of degree 4 and dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
Basis matrix:
[ 1 0 0 0]
[ 0 1 13/(18*d) 103/(432*d^2)]
sage: subspace.ambient_module()
Vector space of dimension 4 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.ambient_module() == MF.module()
True
sage: subspace.ambient_space() == MF
True
sage: subspace.basis()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
sage: subspace.basis()[0].parent() == MF
True
sage: subspace.gens()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
sage: subspace.gens()[0].parent() == subspace
True
sage: subspace.is_ambient()
False
sage: MF = QuasiCuspForms(n=infinity, k=12, ep=1)
sage: MF.dimension()
4
sage: subspace = MF.subspace([MF.Delta(), MF.E4()*MF.f_inf()*MF.E2()*MF.f_i(), MF.E4()*MF.f_inf()*MF.E2()^2, MF.E4()*MF.f_inf()*(MF.E4()-MF.E2()^2)])
sage: subspace.default_prec(3)
sage: subspace
Subspace of dimension 3 of QuasiCuspForms(n=+Infinity, k=12, ep=1) over Integer Ring
sage: subspace.gens()
[q + 24*q^2 + O(q^3), q - 24*q^2 + O(q^3), q - 8*q^2 + O(q^3)]
"""
FormsSpace_abstract.__init__(self, group=ambient_space.group(), base_ring=ambient_space.base_ring(), k=ambient_space.weight(), ep=ambient_space.ep(), n=ambient_space.hecke_n())
Module.__init__(self, base=self.coeff_ring())
self._ambient_space = ambient_space
self._basis = [v for v in basis]
# self(v) instead would somehow mess up the coercion model
self._gens = [self._element_constructor_(v) for v in basis]
self._module = ambient_space._module.submodule([ambient_space.coordinate_vector(v) for v in basis])
# TODO: get the analytic type from the basis
#self._analytic_type=self.AT(["quasi", "mero"])
self._analytic_type = ambient_space._analytic_type