/
subspace.py
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/
subspace.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(ambient_space, basis):
r"""
Return a canonical version of the parameters.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=12, ep=1)
sage: canonical_parameters(MF, [MF.Delta().as_ring_element(), MF.gen(0)])
(ModularForms(n=6, k=12, ep=1) over Integer Ring,
(q + 30*q^2 + 333*q^3 + 1444*q^4 + O(q^5),
1 + 26208*q^3 + 530712*q^4 + O(q^5)))
"""
basis = tuple([ambient_space(v) for v in basis])
# TODO: Check the base ring, reduce vectors to basis, etc
return (ambient_space, basis)
class SubSpaceForms(FormsSpace_abstract, Module, UniqueRepresentation):
r"""
Submodule of (Hecke) forms in the given ambient space for the given basis.
"""
@staticmethod
def __classcall__(cls, ambient_space, basis=()):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=12, ep=1)
sage: (ambient_space, basis) = canonical_parameters(MF, [MF.Delta().as_ring_element(), MF.gen(0)])
sage: SubSpaceForms(MF, [MF.Delta().as_ring_element(), MF.gen(0)]) == SubSpaceForms(ambient_space, basis)
True
"""
(ambient_space, basis) = canonical_parameters(ambient_space, basis)
return super(SubSpaceForms,cls).__classcall__(cls, ambient_space=ambient_space, basis=basis)
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 _repr_(self):
r"""
Return the string representation of ``self``.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=20, ep=1)
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
"""
# If we list the basis the representation usually gets too long...
# return "Subspace with basis {} of {}".format([v.as_ring_element() for v in self.basis()], self._ambient_space)
return "Subspace of dimension {} of {}".format(len(self._basis), self._ambient_space)
def change_ring(self, new_base_ring):
r"""
Return the same space as ``self`` but over a new base ring ``new_base_ring``.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=20, ep=1)
sage: subspace = MF.subspace([MF.Delta()*MF.E4()^2, MF.gen(0)])
sage: subspace.change_ring(CC)
Subspace of dimension 2 of ModularForms(n=6, k=20, ep=1) over Complex Field with 53 bits of precision
"""
return self.__class__.__base__(self._ambient_space.change_ring(new_base_ring), self._basis)
@cached_method
def basis(self):
r"""
Return the basis of ``self`` in the ambient space.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
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
"""
return self._basis
@cached_method
def gens(self):
r"""
Return the basis of ``self``.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
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
"""
return self._gens
@cached_method
def dimension(self):
r"""
Return the dimension of ``self``.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.dimension()
2
sage: subspace.dimension() == len(subspace.gens())
True
"""
return len(self.basis())
@cached_method
def degree(self):
r"""
Return the degree of ``self``.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.degree()
4
sage: subspace.degree() == subspace.ambient_space().degree()
True
"""
return self._ambient_space.degree()
@cached_method
def rank (self):
r"""
Return the rank of ``self``.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.rank()
2
sage: subspace.rank() == subspace.dimension()
True
"""
return len(self.gens())
@cached_method
def coordinate_vector(self, v):
r"""
Return the coordinate vector of ``v`` with respect to
the basis ``self.gens()``.
INPUT:
- ``v``- An element of ``self``.
OUTPUT:
The coordinate vector of ``v`` with respect
to the basis ``self.gens()``.
Note: The coordinate vector is not an element of ``self.module()``.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(group=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.coordinate_vector(MF.gen(0) + MF.Delta()*MF.E4()^2).parent()
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.coordinate_vector(MF.gen(0) + MF.Delta()*MF.E4()^2)
(1, 1)
sage: MF = ModularForms(group=4, k=24, ep=-1)
sage: subspace = MF.subspace([MF.gen(0), MF.gen(2)])
sage: subspace.coordinate_vector(subspace.gen(0)).parent()
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.coordinate_vector(subspace.gen(0))
(1, 0)
"""
return self._module.coordinate_vector(self.ambient_coordinate_vector(v))