Coverage for local/lib/python2.7/site-packages/sage/modular/modform_hecketriangle/subspace.py : 97%

r""" 

Subspaces of modular forms for Hecke triangle groups 

AUTHORS: 

- Jonas Jermann (2013): initial version 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# 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_element import vector 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.misc.cachefunc import cached_method 

from sage.matrix.constructor import matrix 

from .hecke_triangle_groups import HeckeTriangleGroup 

from .abstract_space import FormsSpace_abstract 

def canonical_parameters(ambient_space, basis, check=True): 

r""" 

Return a canonical version of the parameters. 

In particular the list/tuple ``basis`` is replaced by a 

tuple of linearly independent elements in the ambient space. 

If ``check=False`` (default: ``True``) then ``basis`` 

is assumed to already be a basis. 

EXAMPLES:: 

sage: from sage.modular.modform_hecketriangle.subspace import canonical_parameters 

sage: from sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(n=6, k=12, ep=1) 

sage: canonical_parameters(MF, [MF.Delta().as_ring_element(), MF.gen(0), 2*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))) 

""" 

if check: 

coord_matrix = matrix([ambient_space(v).ambient_coordinate_vector() for v in basis]) 

pivots = coord_matrix.transpose().pivots() 

new_basis = [ambient_space(basis[l]) for l in pivots] 

basis = tuple(new_basis) 

else: 

basis = [ambient_space(v) for v in basis] 

basis = tuple(basis) 

return (ambient_space, basis) 

def ModularFormsSubSpace(*args, **kwargs): 

r""" 

Create a modular forms subspace generated by the supplied arguments if possible. 

Instead of a list of generators also multiple input arguments can be used. 

If ``reduce=True`` then the corresponding ambient space is choosen as small as possible. 

If no subspace is available then the ambient space is returned. 

EXAMPLES:: 

sage: from sage.modular.modform_hecketriangle.subspace import ModularFormsSubSpace 

sage: from sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms() 

sage: subspace = ModularFormsSubSpace(MF.E4()^3, MF.E6()^2+MF.Delta(), MF.Delta()) 

sage: subspace 

Subspace of dimension 2 of ModularForms(n=3, k=12, ep=1) over Integer Ring 

sage: subspace.ambient_space() 

ModularForms(n=3, k=12, ep=1) over Integer Ring 

sage: subspace.gens() 

[1 + 720*q + 179280*q^2 + 16954560*q^3 + 396974160*q^4 + O(q^5), 1 - 1007*q + 220728*q^2 + 16519356*q^3 + 399516304*q^4 + O(q^5)] 

sage: ModularFormsSubSpace(MF.E4()^3-MF.E6()^2, reduce=True).ambient_space() 

CuspForms(n=3, k=12, ep=1) over Integer Ring 

sage: ModularFormsSubSpace(MF.E4()^3-MF.E6()^2, MF.J_inv()*MF.E4()^3, reduce=True) 

WeakModularForms(n=3, k=12, ep=1) over Integer Ring 

""" 

generators = [] 

for arg in args: 

if isinstance(arg, list) or isinstance(arg, tuple): 

generators += arg 

else: 

generators.append(arg) 

if ("reduce" in kwargs) and kwargs["reduce"]: 

generators = [gen.full_reduce() for gen in generators] 

if len(generators) == 0: 

raise ValueError("No generators specified") 

el = False 

for gen in generators: 

if el: 

el += gen 

else: 

el = gen 

ambient_space = el.parent() 

try: 

# This works if and only if ambient_space supports subspaces 

ambient_space.coordinate_vector(el) 

generators = [ambient_space(gen) for gen in generators] 

return SubSpaceForms(ambient_space, generators) 

except (NotImplementedError, AttributeError): 

return ambient_space 

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=(), check=True): 

r""" 

Return a (cached) instance with canonical parameters. 

EXAMPLES:: 

sage: from sage.modular.modform_hecketriangle.subspace import (canonical_parameters, SubSpaceForms) 

sage: from sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(n=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, check) 

# we return check=True to ensure only one cached instance 

return super(SubSpaceForms,cls).__classcall__(cls, ambient_space=ambient_space, basis=basis, check=True) 

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 modules over Integer Ring 

sage: subspace in subspace.category() 

True 

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=ambient_space.base_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 sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(n=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 sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(n=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, check=False) 

def change_ambient_space(self, new_ambient_space): 

r""" 

Return a new subspace with the same basis but inside a different ambient space 

(if possible). 

EXAMPLES:: 

sage: from sage.modular.modform_hecketriangle.space import ModularForms, QuasiModularForms 

sage: MF = ModularForms(n=6, k=20, ep=1) 

sage: subspace = MF.subspace([MF.Delta()*MF.E4()^2, MF.gen(0)]) 

sage: new_ambient_space = QuasiModularForms(n=6, k=20, ep=1) 

sage: subspace.change_ambient_space(new_ambient_space) # long time 

Subspace of dimension 2 of QuasiModularForms(n=6, k=20, ep=1) over Integer Ring 

""" 

return self.__class__.__base__(new_ambient_space, self._basis, check=False) 

@cached_method 

def contains_coeff_ring(self): 

r""" 

Return whether ``self`` contains its coefficient ring. 

EXAMPLES:: 

sage: from sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(k=0, ep=1, n=8) 

sage: subspace = MF.subspace([1]) 

sage: subspace.contains_coeff_ring() 

True 

sage: subspace = MF.subspace([]) 

sage: subspace.contains_coeff_ring() 

False 

sage: MF = ModularForms(k=0, ep=-1, n=8) 

sage: subspace = MF.subspace([]) 

sage: subspace.contains_coeff_ring() 

False 

""" 

return (super(SubSpaceForms, self).contains_coeff_ring() and self.dimension()==ZZ(1)) 

@cached_method 

def basis(self): 

r""" 

Return the basis of ``self`` in the ambient space. 

EXAMPLES:: 

sage: from sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(n=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 sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(n=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 sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(n=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 sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(n=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 sage.modular.modform_hecketriangle.space import ModularForms 

sage: MF = ModularForms(n=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 sage.modular.modform_hecketriangle.space import ModularForms, QuasiCuspForms 

sage: MF = ModularForms(n=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(n=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) 

sage: MF = QuasiCuspForms(n=infinity, k=12, ep=1) 

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: el = MF.E4()*MF.f_inf()*(7*MF.E4() - 3*MF.E2()^2) 

sage: subspace.coordinate_vector(el) 

(7, 0, -3) 

sage: subspace.ambient_coordinate_vector(el) 

(7, 21/(8*d), 0, -3) 

""" 

return self._module.coordinate_vector(self.ambient_coordinate_vector(v))