CoCalc -- modularform

Harvard Workshop: Distribution of modular symbols and L-values
code / alex / psage / psage / modform / fourier_expansion_framework / modularforms / modularform_submodule.py
²⁴¹⁸⁵² views
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r"""
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Submodules of rings of orthogonal modular forms.
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AUTHOR :
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    -- Martin Raum (2009 - 07 - 30) Initial version
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"""
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#===============================================================================
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# 
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# Copyright (C) 2009 Martin Raum
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# 
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# This program is free software; you can redistribute it and/or
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# modify it under the terms of the GNU General Public License
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# as published by the Free Software Foundation; either version 3
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# of the License, or (at your option) any later version.
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# 
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# This program is distributed in the hope that it will be useful, 
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# but WITHOUT ANY WARRANTY; without even the implied warranty of
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# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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# General Public License for more details.
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# 
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# You should have received a copy of the GNU General Public License
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# along with this program; if not, see <http://www.gnu.org/licenses/>.
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#
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#===============================================================================
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from psage.modform.fourier_expansion_framework.gradedexpansions.gradedexpansion_ambient import GradedExpansionAmbient_abstract
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from psage.modform.fourier_expansion_framework.gradedexpansions.gradedexpansion_submodule import GradedExpansionSubmodule_ambient_pid, \
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                                                       GradedExpansionSubmodule_submodule_pid, \
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                                                       GradedExpansionSubmodule_abstract, \
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                                                       GradedExpansionSubmoduleVector_generic
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from sage.misc.cachefunc import cached_method
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from sage.structure.sequence import Sequence
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import operator
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#===============================================================================
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# HeckeInvariantSubmodule_abstract
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#===============================================================================
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class HeckeInvariantSubmodule_abstract :       
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    @cached_method
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    def _hecke_action(self, n) :
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        """
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        Calculate the action of `T(n)` on the basis.
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        INPUT:
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            - `n`      -- A Hecke modulus. Probably an integer.
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        OUTPUT:
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            A sequence of elements of self.
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        TESTS::
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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            sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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            sage: sm = ma.graded_submodule(3)
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            sage: sm._hecke_action(7)
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            [(1)]
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        """
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        T = self.graded_ambient().type()._hecke_operator_class()(n)
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        return Sequence( [ self( T.eval(b.fourier_expansion(), self.graded_ambient()(b).weight()) )
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                           for b in self.basis() ],
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                         universe = self )
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    @cached_method
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    def hecke_homomorphism(self, n) :
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        """
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        Calculate the matrix corresponding to the action of `T(n)` on the basis.
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        INPUT:
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            - `n`      -- A Hecke modulus. Probably an integer.
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        OUTPUT:
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            A homomorphism from ``self`` to ``self``.
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        TESTS::
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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            sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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            sage: sm = ma.graded_submodule(3)
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            sage: sm.hecke_homomorphism(7)
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            Free module morphism defined by the matrix
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            [1]
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            Domain: Submodule of Graded expansion module with generators v1, v2, ...
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            Codomain: Submodule of Graded expansion module with generators v1, v2, ...
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        """
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        return self.hom(self._hecke_action(n))
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    @cached_method
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    def hecke_eigenforms(self, n ) :
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        """
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        Return a basis of eigenforms with respect to the Hecke operator `T(n)`.
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        INPUT:
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            - `n`      -- A Hecke modulus. Probably an integer.
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        OUTPUT:
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            A list of elements in the graded ambient.
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        TESTS::
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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            sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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            sage: sm = ma.graded_submodule(3)
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            sage: sm.hecke_eigenforms(7)
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            [Graded expansion v1]
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        """
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        hm = self.hecke_homomorphism(n).matrix().transpose()
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        hes = hm.eigenspaces_right()
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        efs = []
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        for e in hes :
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            ring_elements = Sequence( [sum(map(operator.mul, self._basis_in_graded_ambient(), b.list()))
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                                       for b in e[1].basis() ] )
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            efs += list(ring_elements)
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        return efs 
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#===============================================================================
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# ModularFormsSubmoduleHeckeInvariant
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#===============================================================================
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def ModularFormsSubmoduleHeckeInvariant(arg1, arg2) :
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    """
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    INPUT:
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        - ``arg1``     -- A graded ambient or an ambient module.
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        - ``arg2``     -- A list of elements in ``arg1``. The basis.
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    OUTPUT:
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        An submodule of modular forms.
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    TESTS::
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        sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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        sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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        sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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        sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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        sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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        sage: sm = ma.graded_submodule(6)
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        sage: ssm = ModularFormsSubmoduleHeckeInvariant(sm, [sm.0])
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    """
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    if isinstance(arg1, GradedExpansionAmbient_abstract) :
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        return ModularFormsSubmodule_heckeinvariant_ambient(arg1, arg2)
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    elif isinstance(arg1, GradedExpansionSubmodule_abstract) :
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        return ModularFormsSubmodule_heckeinvariant_submodule(arg1, arg2)
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    else :
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        raise ValueError( "Cannot construct subspace in %s with basis %s." % (arg1, arg2) )
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    return ModularFormsSubmodule_heckeinvariant_ambient
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#===============================================================================
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# ModularFormsSubmodule_heckeinvariant_ambient
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#===============================================================================
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class ModularFormsSubmodule_heckeinvariant_ambient ( 
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        GradedExpansionSubmodule_ambient_pid,
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        HeckeInvariantSubmodule_abstract ) :
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    def __init__(self, graded_ambient, basis, **kwds) :
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        """
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        INPUT:
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            - ``graded_ambient`` -- A graded ambient.
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            - ``basis``          -- A list of elements in the graded ambient.
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        TESTS::
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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            sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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            sage: sm = ModularFormsSubmodule_heckeinvariant_ambient(ma, [ma.0])
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        """
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        if not hasattr(self, '_element_class') :
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            try :
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                _element_class = graded_ambient.type()._space_element_class()
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            except NotImplementedError :
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                _element_class = GradedExpansionSubmoduleVector_generic
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        GradedExpansionSubmodule_ambient_pid.__init__(self, graded_ambient, basis,
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                                                      **kwds)
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#===============================================================================
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# ModularFormsSubmodule_heckeinvariant_submodule
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#===============================================================================
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class ModularFormsSubmodule_heckeinvariant_submodule (
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        GradedExpansionSubmodule_submodule_pid,
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        HeckeInvariantSubmodule_abstract ) :
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    def __init__(self, ambient, basis, **kwds) :
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        """
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        INPUT:
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            - ``ambient`` -- A submodule of modular forms.
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            - ``basis``   -- A list of elements in the ambient.
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        TESTS::
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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            sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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            sage: sm = ModularFormsSubmodule_heckeinvariant_ambient(ma, [ma.0, ma.1])
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            sage: ssm = ModularFormsSubmodule_heckeinvariant_submodule(sm, [sm.1])
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        """
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        if not hasattr(self, '_element_class') :
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            try :
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                _element_class = ambient.graded_ambient().type()._space_element_class()
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            except NotImplementedError :
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                _element_class = GradedExpansionSubmoduleVector_generic
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        self.__integral_basis = None
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        GradedExpansionSubmodule_submodule_pid.__init__(self, ambient, basis,
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                                                        **kwds)
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    def _set_integral_basis(self, basis) :
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        """
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        An additional basis, that will not be echelonized.
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        INPUT:
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            - ``basis`` -- A list of elements of ``self``. A basis for ``self``.
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        OUTPUT:
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            ``None``.
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        TESTS::
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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            sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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            sage: sm = ModularFormsSubmodule_heckeinvariant_ambient(ma, [ma.0, ma.1])
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            sage: ssm = ModularFormsSubmodule_heckeinvariant_submodule(sm, [sm.1])
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            sage: ssm._set_integral_basis([2 * sm.1])
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            sage: ssm._integral_basis()
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            [(0, 2)]
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        """
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        self.__integral_basis = basis
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    def _integral_basis(self) :
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        """
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        An additional basis, that will not be echelonized.
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        OUTPUT:
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            A list of elements of ``self``.
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        TESTS::
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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            sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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            sage: sm = ModularFormsSubmodule_heckeinvariant_ambient(ma, [ma.0, ma.1])
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            sage: ssm = ModularFormsSubmodule_heckeinvariant_submodule(sm, [sm.1])
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            sage: ssm._integral_basis()
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            Traceback (most recent call last):
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            ...
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            RuntimeError: Integral basis is not set.
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        """
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        if self.__integral_basis is None :
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            raise RuntimeError( "Integral basis is not set." )
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        return self.__integral_basis
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#===============================================================================
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# ModularFormsWeightSubmodule
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#===============================================================================
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def ModularFormsWeightSubmodule(graded_ambient, basis, weight) :
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    """
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    A submodule of modular forms, that have the same weights.
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    INPUT:
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        - ``graded_ambient`` -- A graded ambient.
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        - ``basis``          -- A list of elements in the graded ambient.
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        - ``weight``         -- A grading index. The commen weight of all basis elements.
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    OUTPUT:
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        An instance of :class:~`.ModularFormsSubmodule_singleweight_ambient_pid`.
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    TESTS::
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        sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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        sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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        sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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        sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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        sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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        sage: sm = ModularFormsWeightSubmodule(ma, [ma.0], 3)
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    """
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    return ModularFormsSubmodule_singleweight_ambient_pid(graded_ambient, basis, weight)
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#===============================================================================
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# ModularFormSubmodule_singleweight
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#===============================================================================
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class ModularFormsSubmodule_singleweight_ambient_pid ( 
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        GradedExpansionSubmodule_ambient_pid,
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        HeckeInvariantSubmodule_abstract ) :
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    def __init__(self, graded_ambient, basis, weight, **kwds) :
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        """
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        INPUT:
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            - ``graded_ambient``   -- A graded ambient.
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            - ``basis``            -- A list of elements in the graded ambient.
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            - ``weight``           -- A grading index. The commen weight of all basis elements.
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            - ``kwds``             -- Will be forwarded to :class:~`fourier_expansion_framework.gradedexpansions.gradedexpansion_submodule.GradedExpansionSubmodule_ambient_pid`.
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        TESTS::
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_ambient import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
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            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
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            sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_submodule import *
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            sage: ma = ModularFormsModule_withheckeaction( QQ, ModularFormTestType_vectorvalued(), NNFilter(5) )
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            sage: sm = ModularFormsSubmodule_singleweight_ambient_pid(ma, [ma.0], 3)
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        """
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        self.__weight = weight
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        if not hasattr(self, '_element_class') :
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            try :
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                _element_class = graded_ambient.type()._space_element_class()
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            except NotImplementedError :
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                _element_class = GradedExpansionSubmoduleVector_generic
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        GradedExpansionSubmodule_ambient_pid.__init__(self, graded_ambient, basis,
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                                                      **kwds)
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    def weight(self) :
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        return self.__weight
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## TODO: Provide standard implementation of is_cusp_form for vectors
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## TODO: Provide implementation of homogeneous weight vectors and
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##       make Hecke operators work for this
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