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r""" Graded rings of 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/ #*****************************************************************************
r""" Return a canonical version of the parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import canonical_parameters sage: canonical_parameters(4, ZZ, 1) (Hecke triangle group for n = 4, Integer Ring, True, 4) sage: canonical_parameters(infinity, RR, 0) (Hecke triangle group for n = +Infinity, Real Field with 53 bits of precision, False, +Infinity) """
else:
r""" Graded ring of (Hecke) quasi meromorphic modular forms for the given group and base ring. """
r""" Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, QuasiMeromorphicModularFormsRing) sage: (group, base_ring, red_hom, n) = canonical_parameters(4, ZZ, 1) sage: QuasiMeromorphicModularFormsRing(4, ZZ, 1) == QuasiMeromorphicModularFormsRing(group, base_ring, red_hom, n) True """
r""" Return the graded ring of (Hecke) quasi meromorphic modular forms for the given ``group`` and ``base_ring``.
INPUT:
- ``group`` -- The Hecke triangle group (default: ``HeckeTriangleGroup(3)``)
- ``base_ring`` -- The base_ring (default: ``ZZ``).
- ``red_hom`` -- If True then results of binary operations are considered homogeneous whenever it makes sense (default: False). This is mainly used by the spaces of homogeneous elements.
OUTPUT:
The corresponding graded ring of (Hecke) quasi meromorphic modular forms for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing sage: MR = QuasiMeromorphicModularFormsRing(4, ZZ, 1) sage: MR QuasiMeromorphicModularFormsRing(n=4) over Integer Ring sage: MR.analytic_type() quasi meromorphic modular sage: MR.category() Category of commutative algebras over Integer Ring sage: MR in MR.category() True
sage: QuasiMeromorphicModularFormsRing(n=infinity) QuasiMeromorphicModularFormsRing(n=+Infinity) over Integer Ring """
r""" Graded ring of (Hecke) quasi weakly holomorphic modular forms for the given group and base ring. """
r""" Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, QuasiWeakModularFormsRing) sage: (group, base_ring, red_hom, n) = canonical_parameters(5, CC, 0) sage: QuasiWeakModularFormsRing(5, CC, 0) == QuasiWeakModularFormsRing(group, base_ring, red_hom, n) True """
r""" Return the graded ring of (Hecke) quasi weakly holomorphic modular forms for the given ``group`` and ``base_ring``.
INPUT:
- ``group`` -- The Hecke triangle group (default: ``HeckeTriangleGroup(3)``)
- ``base_ring`` -- The base_ring (default: ``ZZ``).
- ``red_hom`` -- If True then results of binary operations are considered homogeneous whenever it makes sense (default: False). This is mainly used by the spaces of homogeneous elements.
OUTPUT:
The corresponding graded ring of (Hecke) quasi weakly holomorphic modular forms for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiWeakModularFormsRing sage: MR = QuasiWeakModularFormsRing(5, CC, 0) sage: MR QuasiWeakModularFormsRing(n=5) over Complex Field with 53 bits of precision sage: MR.analytic_type() quasi weakly holomorphic modular sage: MR.category() Category of commutative algebras over Complex Field with 53 bits of precision sage: MR in MR.category() True """
r""" Graded ring of (Hecke) quasi modular forms for the given group and base ring """
r""" Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, QuasiModularFormsRing) sage: (group, base_ring, red_hom, n) = canonical_parameters(6, ZZ, True) sage: QuasiModularFormsRing(6, ZZ, True) == QuasiModularFormsRing(group, base_ring, red_hom, n) True """
r""" Return the graded ring of (Hecke) quasi modular forms for the given ``group`` and ``base_ring``.
INPUT:
- ``group`` -- The Hecke triangle group (default: ``HeckeTriangleGroup(3)``)
- ``base_ring`` -- The base_ring (default: ``ZZ``).
- ``red_hom`` -- If True then results of binary operations are considered homogeneous whenever it makes sense (default: False). This is mainly used by the spaces of homogeneous elements.
OUTPUT:
The corresponding graded ring of (Hecke) quasi modular forms for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing sage: MR = QuasiModularFormsRing(6, ZZ, True) sage: MR QuasiModularFormsRing(n=6) over Integer Ring sage: MR.analytic_type() quasi modular sage: MR.category() Category of commutative algebras over Integer Ring sage: MR in MR.category() True """
r""" Graded ring of (Hecke) quasi cusp forms for the given group and base ring. """
r""" Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, QuasiCuspFormsRing) sage: (group, base_ring, red_hom, n) = canonical_parameters(7, ZZ, 1) sage: QuasiCuspFormsRing(7, ZZ, 1) == QuasiCuspFormsRing(group, base_ring, red_hom, n) True """
r""" Return the graded ring of (Hecke) quasi cusp forms for the given ``group`` and ``base_ring``.
INPUT:
- ``group`` -- The Hecke triangle group (default: ``HeckeTriangleGroup(3)``)
- ``base_ring`` -- The base_ring (default: ``ZZ``).
- ``red_hom`` -- If True then results of binary operations are considered homogeneous whenever it makes sense (default: False). This is mainly used by the spaces of homogeneous elements.
OUTPUT:
The corresponding graded ring of (Hecke) quasi cusp forms for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiCuspFormsRing sage: MR = QuasiCuspFormsRing(7, ZZ, 1) sage: MR QuasiCuspFormsRing(n=7) over Integer Ring sage: MR.analytic_type() quasi cuspidal sage: MR.category() Category of commutative algebras over Integer Ring sage: MR in MR.category() True """
r""" Graded ring of (Hecke) meromorphic modular forms for the given group and base ring """
r""" Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, MeromorphicModularFormsRing) sage: (group, base_ring, red_hom, n) = canonical_parameters(4, ZZ, 1) sage: MeromorphicModularFormsRing(4, ZZ, 1) == MeromorphicModularFormsRing(group, base_ring, red_hom, n) True """
r""" Return the graded ring of (Hecke) meromorphic modular forms for the given ``group`` and ``base_ring``.
INPUT:
- ``group`` -- The Hecke triangle group (default: ``HeckeTriangleGroup(3)``)
- ``base_ring`` -- The base_ring (default: ``ZZ``).
- ``red_hom`` -- If True then results of binary operations are considered homogeneous whenever it makes sense (default: False). This is mainly used by the spaces of homogeneous elements.
OUTPUT:
The corresponding graded ring of (Hecke) meromorphic modular forms for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import MeromorphicModularFormsRing sage: MR = MeromorphicModularFormsRing(4, ZZ, 1) sage: MR MeromorphicModularFormsRing(n=4) over Integer Ring sage: MR.analytic_type() meromorphic modular sage: MR.category() Category of commutative algebras over Integer Ring sage: MR in MR.category() True """
r""" Graded ring of (Hecke) weakly holomorphic modular forms for the given group and base ring """
r""" Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, WeakModularFormsRing) sage: (group, base_ring, red_hom, n) = canonical_parameters(5, ZZ, 0) sage: WeakModularFormsRing(5, ZZ, 0) == WeakModularFormsRing(group, base_ring, red_hom, n) True """
r""" Return the graded ring of (Hecke) weakly holomorphic modular forms for the given ``group`` and ``base_ring``.
INPUT:
- ``group`` -- The Hecke triangle group (default: ``HeckeTriangleGroup(3)``)
- ``base_ring`` -- The base_ring (default: ``ZZ``).
- ``red_hom`` -- If True then results of binary operations are considered homogeneous whenever it makes sense (default: False). This is mainly used by the spaces of homogeneous elements.
OUTPUT:
The corresponding graded ring of (Hecke) weakly holomorphic modular forms for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing sage: MR = WeakModularFormsRing(5, ZZ, 0) sage: MR WeakModularFormsRing(n=5) over Integer Ring sage: MR.analytic_type() weakly holomorphic modular sage: MR.category() Category of commutative algebras over Integer Ring sage: MR in MR.category() True """
r""" Graded ring of (Hecke) modular forms for the given group and base ring """
r""" Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing sage: ModularFormsRing(3, ZZ, 0) == ModularFormsRing() True """
r""" Return the graded ring of (Hecke) modular forms for the given ``group`` and ``base_ring``.
INPUT:
- ``group`` -- The Hecke triangle group (default: ``HeckeTriangleGroup(3)``)
- ``base_ring`` -- The base_ring (default: ``ZZ``).
- ``red_hom`` -- If True then results of binary operations are considered homogeneous whenever it makes sense (default: False). This is mainly used by the spaces of homogeneous elements.
OUTPUT:
The corresponding graded ring of (Hecke) modular forms for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing sage: MR = ModularFormsRing() sage: MR ModularFormsRing(n=3) over Integer Ring sage: MR.analytic_type() modular sage: MR.category() Category of commutative algebras over Integer Ring sage: MR in MR.category() True """
r""" Graded ring of (Hecke) cusp forms for the given group and base ring """
r""" Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, CuspFormsRing) sage: (group, base_ring, red_hom, n) = canonical_parameters(5, CC, True) sage: CuspFormsRing(5, CC, True) == CuspFormsRing(group, base_ring, red_hom, n) True """
r""" Return the graded ring of (Hecke) cusp forms for the given ``group`` and ``base_ring``.
INPUT:
- ``group`` -- The Hecke triangle group (default: ``HeckeTriangleGroup(3)``)
- ``base_ring`` -- The base_ring (default: ``ZZ``).
- ``red_hom`` -- If True then results of binary operations are considered homogeneous whenever it makes sense (default: False). This is mainly used by the spaces of homogeneous elements.
OUTPUT:
The corresponding graded ring of (Hecke) cusp forms for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.graded_ring import CuspFormsRing sage: MR = CuspFormsRing(5, CC, True) sage: MR CuspFormsRing(n=5) over Complex Field with 53 bits of precision sage: MR.analytic_type() cuspidal sage: MR.category() Category of commutative algebras over Complex Field with 53 bits of precision sage: MR in MR.category() True
sage: CuspFormsRing(n=infinity, base_ring=CC, red_hom=True) CuspFormsRing(n=+Infinity) over Complex Field with 53 bits of precision """
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