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r""" 

Graded rings 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 FractionField, PolynomialRing, PowerSeriesRing, ZZ, QQ, infinity 

from sage.algebras.free_algebra import FreeAlgebra 

 

from sage.structure.parent import Parent 

from sage.misc.cachefunc import cached_method 

 

from .constructor import FormsRing, FormsSpace 

from .series_constructor import MFSeriesConstructor 

 

 

# Maybe replace Parent by just SageObject? 

class FormsRing_abstract(Parent): 

r""" 

Abstract (Hecke) forms ring. 

 

This should never be called directly. Instead one should 

instantiate one of the derived classes of this class. 

""" 

 

from .graded_ring_element import FormsRingElement 

Element = FormsRingElement 

 

from .analytic_type import AnalyticType 

AT = AnalyticType() 

 

def __init__(self, group, base_ring, red_hom, n): 

r""" 

Abstract (Hecke) forms ring. 

 

INPUT: 

 

- ``group`` -- The Hecke triangle group (default: ``HeckeTriangleGroup(3)``) 

 

- ``base_ring`` -- The base_ring (default: `\Z). 

 

- ``red_hom`` -- If ``True`` then results of binary operations are considered 

homogeneous whenever it makes sense (default: ``False``). 

This is mainly used by the (Hecke) forms. 

 

OUTPUT: 

 

The corresponding abstract (Hecke) forms ring. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: MR = ModularFormsRing(n=5, base_ring=ZZ, red_hom=True) 

sage: MR 

ModularFormsRing(n=5) over Integer Ring 

sage: MR.group() 

Hecke triangle group for n = 5 

sage: MR.base_ring() 

Integer Ring 

sage: MR.has_reduce_hom() 

True 

sage: MR.is_homogeneous() 

False 

""" 

 

#from graded_ring import canonical_parameters 

#(group, base_ring, red_hom, n) = canonical_parameters(group, base_ring, red_hom, n) 

 

#if (not group.is_arithmetic() and base_ring.characteristic()>0): 

# raise NotImplementedError 

#if (base_ring.characteristic().divides(2*group.n()*(group.n()-2))): 

# raise NotImplementedError 

 

if (base_ring.characteristic() > 0): 

raise NotImplementedError("only characteristic 0 is supported") 

self._group = group 

self._red_hom = red_hom 

self._base_ring = base_ring 

self._coeff_ring = FractionField(PolynomialRing(base_ring,'d')) 

self._pol_ring = PolynomialRing(base_ring,'x,y,z,d') 

self._rat_field = FractionField(self._pol_ring) 

 

# default values 

self._weight = None 

self._ep = None 

self._analytic_type = self.AT(["quasi", "mero"]) 

 

self.default_prec(10) 

self.disp_prec(5) 

self.default_num_prec(53) 

 

#super(FormsRing_abstract, self).__init__(self.coeff_ring()) 

 

def _repr_(self): 

r""" 

Return the string representation of ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing 

sage: QuasiModularFormsRing(n=4) 

QuasiModularFormsRing(n=4) over Integer Ring 

""" 

 

return "{}FormsRing(n={}) over {}".format(self._analytic_type.analytic_space_name(), self._group.n(), self._base_ring) 

 

def _latex_(self): 

r""" 

Return the LaTeX representation of ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiWeakModularFormsRing 

sage: latex(QuasiWeakModularFormsRing()) 

\mathcal{ QM^! }_{n=3}(\Bold{Z}) 

""" 

 

from sage.misc.latex import latex 

return "\\mathcal{{ {} }}_{{n={}}}({})".format(self._analytic_type.latex_space_name(), self._group.n(), latex(self._base_ring)) 

 

def _element_constructor_(self, el): 

r""" 

Return ``el`` coerced/converted into this forms ring. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: MR = ModularFormsRing() 

sage: (x,y,z,d) = MR.pol_ring().gens() 

 

sage: MR(x^3) 

f_rho^3 

 

sage: el = MR.Delta().full_reduce() 

sage: MR(el) 

f_rho^3*d - f_i^2*d 

sage: el.parent() == MR 

False 

sage: MR(el).parent() == MR 

True 

 

sage: el = MR.Delta().full_reduce() 

sage: MRinf = ModularFormsRing(n=infinity) 

sage: MRinf(el) 

(E4*f_i^4 - 2*E4^2*f_i^2 + E4^3)/4096 

sage: el.parent() 

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

sage: MRinf(el).parent() 

ModularFormsRing(n=+Infinity) over Integer Ring 

""" 

 

from .graded_ring_element import FormsRingElement 

if isinstance(el, FormsRingElement): 

if (self.hecke_n() == infinity and el.hecke_n() == ZZ(3)): 

el_f = el._reduce_d()._rat 

(x,y,z,d) = self.pol_ring().gens() 

 

num_sub = el_f.numerator().subs( x=(y**2 + 3*x)/ZZ(4), y=(9*x*y - y**3)/ZZ(8), z=(3*z - y)/ZZ(2)) 

denom_sub = el_f.denominator().subs( x=(y**2 + 3*x)/ZZ(4), y=(9*x*y - y**3)/ZZ(8), z=(3*z - y)/ZZ(2)) 

new_num = num_sub.numerator()*denom_sub.denominator() 

new_denom = denom_sub.numerator()*num_sub.denominator() 

 

el = self._rat_field(new_num) / self._rat_field(new_denom) 

elif self.group() == el.group(): 

el = self._rat_field(el._rat) 

else: 

raise ValueError("{} has group {} != {}".format(el, el.group(), self.group())) 

else: 

el = self._rat_field(el) 

return self.element_class(self, el) 

 

def _coerce_map_from_(self, S): 

r""" 

Return whether or not there exists a coercion from ``S`` to ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiWeakModularFormsRing, ModularFormsRing, CuspFormsRing 

sage: MR1 = QuasiWeakModularFormsRing(base_ring=CC) 

sage: MR2 = ModularFormsRing() 

sage: MR3 = CuspFormsRing() 

sage: MR4 = ModularFormsRing(n=infinity) 

sage: MR5 = ModularFormsRing(n=4) 

sage: MR3.has_coerce_map_from(MR2) 

False 

sage: MR1.has_coerce_map_from(MR2) 

True 

sage: MR2.has_coerce_map_from(MR3) 

True 

sage: MR3.has_coerce_map_from(ZZ) 

False 

sage: MR1.has_coerce_map_from(ZZ) 

True 

sage: MR4.has_coerce_map_from(MR2) 

True 

sage: MR4.has_coerce_map_from(MR5) 

False 

 

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

sage: MF2 = ModularForms(k=6, ep=-1) 

sage: MF3 = CuspForms(k=12, ep=1) 

sage: MR1.has_coerce_map_from(MF2) 

True 

sage: MR2.has_coerce_map_from(MF3) 

True 

sage: MR4.has_coerce_map_from(MF2) 

True 

""" 

 

from .space import FormsSpace_abstract 

from .functors import _common_subgroup 

if ( isinstance(S, FormsRing_abstract)\ 

and self._group == _common_subgroup(self._group, S._group)\ 

and self._analytic_type >= S._analytic_type\ 

and self.base_ring().has_coerce_map_from(S.base_ring()) ): 

return True 

elif isinstance(S, FormsRing_abstract): 

return False 

elif isinstance(S, FormsSpace_abstract): 

raise RuntimeError( "This case should not occur." ) 

# return self._coerce_map_from_(S.graded_ring()) 

elif (self.AT("holo") <= self._analytic_type) and (self.coeff_ring().has_coerce_map_from(S)): 

return True 

else: 

return False 

 

def _an_element_(self): 

r""" 

Return an element of ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import CuspFormsRing 

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

sage: CuspFormsRing().an_element() 

f_rho^3*d - f_i^2*d 

sage: CuspFormsRing().an_element() == CuspFormsRing().Delta() 

True 

sage: WeakModularForms().an_element() 

O(q^5) 

sage: WeakModularForms().an_element() == WeakModularForms().zero() 

True 

""" 

 

return self(self.Delta()) 

 

def default_prec(self, prec = None): 

r""" 

Set the default precision ``prec`` for the Fourier expansion. 

If ``prec=None`` (default) then the current default precision is returned instead. 

 

INPUT: 

 

- ``prec`` -- An integer. 

 

NOTE: 

 

This is also used as the default precision for the Fourier 

expansion when evaluating forms. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

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

sage: MR = ModularFormsRing() 

sage: MR.default_prec(3) 

sage: MR.default_prec() 

3 

sage: MR.Delta().q_expansion_fixed_d() 

q - 24*q^2 + O(q^3) 

sage: MF = ModularForms(k=4) 

sage: MF.default_prec(2) 

sage: MF.E4() 

1 + 240*q + O(q^2) 

sage: MF.default_prec() 

2 

""" 

 

if (prec is not None): 

self._prec = ZZ(prec) 

else: 

return self._prec 

 

def disp_prec(self, prec = None): 

r""" 

Set the maximal display precision to ``prec``. 

If ``prec="max"`` the precision is set to the default precision. 

If ``prec=None`` (default) then the current display precision is returned instead. 

 

NOTE: 

 

This is used for displaying/representing (elements of) 

``self`` as Fourier expansions. 

 

EXAMPLES:: 

 

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

sage: MF = ModularForms(k=4) 

sage: MF.default_prec(5) 

sage: MF.disp_prec(3) 

sage: MF.disp_prec() 

3 

sage: MF.E4() 

1 + 240*q + 2160*q^2 + O(q^3) 

sage: MF.disp_prec("max") 

sage: MF.E4() 

1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + O(q^5) 

""" 

 

if (prec == "max"): 

self._disp_prec = self._prec; 

elif (prec is not None): 

self._disp_prec = ZZ(prec) 

else: 

return self._disp_prec 

 

def default_num_prec(self, prec = None): 

r""" 

Set the default numerical precision to ``prec`` (default: ``53``). 

If ``prec=None`` (default) the current default numerical 

precision is returned instead. 

 

EXAMPLES:: 

 

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

sage: MF = ModularForms(k=6) 

sage: MF.default_prec(20) 

sage: MF.default_num_prec(10) 

sage: MF.default_num_prec() 

10 

sage: E6 = MF.E6() 

sage: E6(i + 1e-1000) 

0.002... - 6.7...e-1000*I 

sage: MF.default_num_prec(100) 

sage: E6(i + 1e-1000) 

3.9946838...e-1999 - 6.6578064...e-1000*I 

 

sage: MF = ModularForms(n=5, k=4/3) 

sage: f_rho = MF.f_rho() 

sage: f_rho.q_expansion(prec=2)[1] 

7/(100*d) 

sage: MF.default_num_prec(15) 

sage: f_rho.q_expansion_fixed_d(prec=2)[1] 

9.9... 

sage: MF.default_num_prec(100) 

sage: f_rho.q_expansion_fixed_d(prec=2)[1] 

9.92593243510795915276017782... 

""" 

 

if (prec is not None): 

self._num_prec = ZZ(prec) 

else: 

return self._num_prec 

 

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.graded_ring import ModularFormsRing 

sage: ModularFormsRing().change_ring(CC) 

ModularFormsRing(n=3) over Complex Field with 53 bits of precision 

""" 

 

return self.__class__.__base__(self._group, new_base_ring, self._red_hom) 

 

def graded_ring(self): 

r""" 

Return the graded ring containing ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing, CuspFormsRing 

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

 

sage: MR = ModularFormsRing(n=5) 

sage: MR.graded_ring() == MR 

True 

 

sage: CF=CuspForms(k=12) 

sage: CF.graded_ring() == CuspFormsRing() 

False 

sage: CF.graded_ring() == CuspFormsRing(red_hom=True) 

True 

 

sage: CF.subspace([CF.Delta()]).graded_ring() == CuspFormsRing(red_hom=True) 

True 

""" 

 

return self.extend_type(ring=True) 

 

def extend_type(self, analytic_type=None, ring=False): 

r""" 

Return a new space which contains (elements of) ``self`` with the analytic type 

of ``self`` extended by ``analytic_type``, possibly extended to a graded ring 

in case ``ring`` is ``True``. 

 

INPUT: 

 

- ``analytic_type`` -- An ``AnalyticType`` or something which 

coerces into it (default: ``None``). 

 

- ``ring`` -- Whether to extend to a graded ring (default: ``False``). 

 

OUTPUT: 

 

The new extended space. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

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

 

sage: MR = ModularFormsRing(n=5) 

sage: MR.extend_type(["quasi", "weak"]) 

QuasiWeakModularFormsRing(n=5) over Integer Ring 

 

sage: CF=CuspForms(k=12) 

sage: CF.extend_type("holo") 

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

sage: CF.extend_type("quasi", ring=True) 

QuasiCuspFormsRing(n=3) over Integer Ring 

 

sage: CF.subspace([CF.Delta()]).extend_type() 

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

""" 

 

if analytic_type is None: 

analytic_type = self._analytic_type 

else: 

analytic_type = self._analytic_type.extend_by(analytic_type) 

 

if (ring or not self.is_homogeneous()): 

return FormsRing(analytic_type, group=self.group(), base_ring=self.base_ring(), red_hom=self.has_reduce_hom()) 

else: 

return FormsSpace(analytic_type, group=self.group(), base_ring=self.base_ring(), k=self.weight(), ep=self.ep()) 

 

def reduce_type(self, analytic_type=None, degree=None): 

r""" 

Return a new space with analytic properties shared by both ``self`` and ``analytic_type``, 

possibly reduced to its space of homogeneous elements of the given ``degree`` (if ``degree`` is set). 

Elements of the new space are contained in ``self``. 

 

INPUT: 

 

- ``analytic_type`` -- An ``AnalyticType`` or something which coerces into it (default: ``None``). 

 

- ``degree`` -- ``None`` (default) or the degree of the homogeneous component to which 

``self`` should be reduced. 

 

OUTPUT: 

 

The new reduced space. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing 

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

 

sage: MR = QuasiModularFormsRing() 

sage: MR.reduce_type(["quasi", "cusp"]) 

QuasiCuspFormsRing(n=3) over Integer Ring 

 

sage: MR.reduce_type("cusp", degree=(12,1)) 

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

 

sage: MF=QuasiModularForms(k=6) 

sage: MF.reduce_type("holo") 

ModularForms(n=3, k=6, ep=-1) over Integer Ring 

 

sage: MF.reduce_type([]) 

ZeroForms(n=3, k=6, ep=-1) over Integer Ring 

""" 

 

if analytic_type is None: 

analytic_type = self._analytic_type 

else: 

analytic_type = self._analytic_type.reduce_to(analytic_type) 

 

if (degree is None and not self.is_homogeneous()): 

return FormsRing(analytic_type, group=self.group(), base_ring=self.base_ring(), red_hom=self.has_reduce_hom()) 

elif (degree is None): 

return FormsSpace(analytic_type, group=self.group(), base_ring=self.base_ring(), k=self.weight(), ep=self.ep()) 

else: 

(weight, ep) = degree 

if (self.is_homogeneous() and (weight != self.weight() or ep!=self.ep())): 

analytic_type = self._analytic_type.reduce_to([]) 

return FormsSpace(analytic_type, group=self.group(), base_ring=self.base_ring(), k=weight, ep=ep) 

 

@cached_method 

def contains_coeff_ring(self): 

r""" 

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

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import CuspFormsRing, ModularFormsRing 

sage: CuspFormsRing(n=4).contains_coeff_ring() 

False 

sage: ModularFormsRing(n=5).contains_coeff_ring() 

True 

""" 

 

return (self.AT("holo") <= self._analytic_type) 

 

def construction(self): 

r""" 

Return a functor that constructs ``self`` (used by the coercion machinery). 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing().construction() 

(ModularFormsRingFunctor(n=3), BaseFacade(Integer Ring)) 

""" 

 

from .functors import FormsRingFunctor, BaseFacade 

return FormsRingFunctor(self._analytic_type, self._group, self._red_hom), BaseFacade(self._base_ring) 

 

@cached_method 

def group(self): 

r""" 

Return the (Hecke triangle) group of ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: MR = ModularFormsRing(n=7) 

sage: MR.group() 

Hecke triangle group for n = 7 

 

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

sage: CF = CuspForms(n=7, k=4/5) 

sage: CF.group() 

Hecke triangle group for n = 7 

""" 

 

return self._group 

 

@cached_method 

def hecke_n(self): 

r""" 

Return the parameter ``n`` of the 

(Hecke triangle) group of ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: MR = ModularFormsRing(n=7) 

sage: MR.hecke_n() 

7 

 

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

sage: CF = CuspForms(n=7, k=4/5) 

sage: CF.hecke_n() 

7 

""" 

 

return self._group.n() 

 

@cached_method 

def base_ring(self): 

r""" 

Return base ring of ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing().base_ring() 

Integer Ring 

 

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

sage: CuspForms(k=12, base_ring=AA).base_ring() 

Algebraic Real Field 

""" 

 

return self._base_ring 

 

@cached_method 

def coeff_ring(self): 

r""" 

Return coefficient ring of ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing().coeff_ring() 

Fraction Field of Univariate Polynomial Ring in d over Integer Ring 

 

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

sage: CuspForms(k=12, base_ring=AA).coeff_ring() 

Fraction Field of Univariate Polynomial Ring in d over Algebraic Real Field 

""" 

 

return self._coeff_ring 

 

@cached_method 

def pol_ring(self): 

r""" 

Return the underlying polynomial ring used 

by ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing().pol_ring() 

Multivariate Polynomial Ring in x, y, z, d over Integer Ring 

 

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

sage: CuspForms(k=12, base_ring=AA).pol_ring() 

Multivariate Polynomial Ring in x, y, z, d over Algebraic Real Field 

""" 

 

return self._pol_ring 

 

@cached_method 

def rat_field(self): 

r""" 

Return the underlying rational field used by 

``self`` to construct/represent elements. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing().rat_field() 

Fraction Field of Multivariate Polynomial Ring in x, y, z, d over Integer Ring 

 

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

sage: CuspForms(k=12, base_ring=AA).rat_field() 

Fraction Field of Multivariate Polynomial Ring in x, y, z, d over Algebraic Real Field 

""" 

 

return self._rat_field 

 

def get_d(self, fix_d = False, d_num_prec = None): 

r""" 

Return the parameter ``d`` of self either as a formal 

parameter or as a numerical approximation with the specified 

precision (resp. an exact value in the arithmetic cases). 

 

For an (exact) symbolic expression also see 

``HeckeTriangleGroup().dvalue()``. 

 

INPUT: 

 

- ``fix_d`` -- If ``False`` (default) a formal parameter is 

used for ``d``. 

 

If ``True`` then the numerical value of 

``d`` is used (or an exact value if the 

group is arithmetic). Otherwise, the given 

value is used for ``d``. 

 

- ``d_num_prec`` -- An integer. The numerical precision of 

``d``. Default: ``None``, in which case 

the default numerical precision of 

``self.parent()`` is used. 

 

OUTPUT: 

 

The corresponding formal, numerical or exact parameter ``d`` of ``self``, 

depending on the arguments and whether ``self.group()`` is arithmetic. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing(n=8).get_d() 

d 

sage: ModularFormsRing(n=8).get_d().parent() 

Fraction Field of Univariate Polynomial Ring in d over Integer Ring 

sage: ModularFormsRing(n=infinity).get_d(fix_d = True) 

1/64 

sage: ModularFormsRing(n=infinity).get_d(fix_d = True).parent() 

Rational Field 

sage: ModularFormsRing(n=5).default_num_prec(40) 

sage: ModularFormsRing(n=5).get_d(fix_d = True) 

0.0070522341... 

sage: ModularFormsRing(n=5).get_d(fix_d = True).parent() 

Real Field with 40 bits of precision 

sage: ModularFormsRing(n=5).get_d(fix_d = True, d_num_prec=100).parent() 

Real Field with 100 bits of precision 

sage: ModularFormsRing(n=5).get_d(fix_d=1).parent() 

Integer Ring 

""" 

 

if d_num_prec is None: 

d_num_prec = self.default_num_prec() 

else: 

d_num_prec = ZZ(d_num_prec) 

 

if (fix_d is True): 

d = self._group.dvalue() 

if (self._group.is_arithmetic()): 

d = 1 / self.base_ring()(1/d) 

else: 

d = self.group().dvalue().n(d_num_prec) 

elif (fix_d is False): 

d = FractionField(PolynomialRing(self.base_ring(), "d")).gen() 

else: 

d = fix_d 

 

return d 

 

def get_q(self, prec = None, fix_d = False, d_num_prec = None): 

r""" 

Return the generator of the power series of the Fourier expansion of ``self``. 

 

INPUT: 

 

- ``prec`` -- An integer or ``None`` (default), namely the desired default 

precision of the space of power series. If nothing is specified 

the default precision of ``self`` is used. 

 

- ``fix_d`` -- If ``False`` (default) a formal parameter is used for ``d``. 

If ``True`` then the numerical value of ``d`` is used 

(resp. an exact value if the group is arithmetic). 

Otherwise the given value is used for ``d``. 

 

- ``d_num_prec`` -- The precision to be used if a numerical value for ``d`` is substituted. 

Default: ``None`` in which case the default 

numerical precision of ``self.parent()`` is used. 

 

OUTPUT: 

 

The generator of the ``PowerSeriesRing`` of corresponding to the given 

parameters. The base ring of the power series ring is given by the corresponding 

parent of ``self.get_d()`` with the same arguments. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing(n=8).default_prec(5) 

sage: ModularFormsRing(n=8).get_q().parent() 

Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring 

sage: ModularFormsRing(n=8).get_q().parent().default_prec() 

5 

sage: ModularFormsRing(n=infinity).get_q(prec=12, fix_d = True).parent() 

Power Series Ring in q over Rational Field 

sage: ModularFormsRing(n=infinity).get_q(prec=12, fix_d = True).parent().default_prec() 

12 

sage: ModularFormsRing(n=5).default_num_prec(40) 

sage: ModularFormsRing(n=5).get_q(fix_d = True).parent() 

Power Series Ring in q over Real Field with 40 bits of precision 

sage: ModularFormsRing(n=5).get_q(fix_d = True, d_num_prec=100).parent() 

Power Series Ring in q over Real Field with 100 bits of precision 

sage: ModularFormsRing(n=5).get_q(fix_d=1).parent() 

Power Series Ring in q over Rational Field 

""" 

 

d = self.get_d(fix_d, d_num_prec) 

if (prec is None): 

prec = self.default_prec() 

 

base_ring = d.parent() 

return PowerSeriesRing(FractionField(base_ring), 'q', default_prec = prec).gen() 

 

@cached_method 

def diff_alg(self): 

r""" 

Return the algebra of differential operators 

(over QQ) which is used on rational functions 

representing elements of ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing().diff_alg() 

Noncommutative Multivariate Polynomial Ring in X, Y, Z, dX, dY, dZ over Rational Field, nc-relations: {dZ*Z: Z*dZ + 1, dY*Y: Y*dY + 1, dX*X: X*dX + 1} 

 

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

sage: CuspForms(k=12, base_ring=AA).diff_alg() 

Noncommutative Multivariate Polynomial Ring in X, Y, Z, dX, dY, dZ over Rational Field, nc-relations: {dZ*Z: Z*dZ + 1, dY*Y: Y*dY + 1, dX*X: X*dX + 1} 

""" 

 

# We only use two operators for now which do not involve 'd', so for performance 

# reason and due to restrictions for possible rings that can be used with algebra 

# relations we choose FractionField(base_ring) instead of self.coeff_ring(). 

# For our purposes it is currently enough to define the operators over ZZ resp. QQ. 

free_alg = FreeAlgebra(FractionField(ZZ),6,'X,Y,Z,dX,dY,dZ') 

(X,Y,Z,dX,dY,dZ) = free_alg.gens() 

diff_alg = free_alg.g_algebra({dX*X:1+X*dX,dY*Y:1+Y*dY,dZ*Z:1+Z*dZ}) 

 

return diff_alg 

 

@cached_method 

def _derivative_op(self): 

r""" 

Return the differential operator in ``self.diff_alg()`` 

corresponding to the derivative of forms. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing(n=7)._derivative_op() 

-1/2*X^6*dY - 5/28*X^5*dZ + 1/7*X*Z*dX + 1/2*Y*Z*dY + 5/28*Z^2*dZ - 1/7*Y*dX 

 

sage: ModularFormsRing(n=infinity)._derivative_op() 

-X*Y*dX + X*Z*dX + 1/2*Y*Z*dY + 1/4*Z^2*dZ - 1/2*X*dY - 1/4*X*dZ 

""" 

 

(X,Y,Z,dX,dY,dZ) = self.diff_alg().gens() 

 

if (self.hecke_n() == infinity): 

return (X*Z-X*Y) * dX\ 

+ ZZ(1)/ZZ(2) * (Y*Z-X) * dY\ 

+ ZZ(1)/ZZ(4) * (Z**2-X) * dZ 

else: 

return 1/self._group.n() * (X*Z-Y) * dX\ 

+ ZZ(1)/ZZ(2) * (Y*Z-X**(self._group.n()-1)) * dY\ 

+ (self._group.n()-2) / (4*self._group.n()) * (Z**2-X**(self._group.n()-2)) * dZ 

 

@cached_method 

def _serre_derivative_op(self): 

r""" 

Return the differential operator in ``self.diff_alg()`` 

corresponding to the Serre derivative of forms. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing(n=8)._serre_derivative_op() 

-1/2*X^7*dY - 3/16*X^6*dZ - 3/16*Z^2*dZ - 1/8*Y*dX 

 

sage: ModularFormsRing(n=infinity)._serre_derivative_op() 

-X*Y*dX - 1/4*Z^2*dZ - 1/2*X*dY - 1/4*X*dZ 

""" 

 

(X,Y,Z,dX,dY,dZ) = self.diff_alg().gens() 

 

if (self.hecke_n() == infinity): 

return - X * Y * dX\ 

- ZZ(1)/ZZ(2) * X * dY\ 

- ZZ(1)/ZZ(4) * (Z**2+X) * dZ 

else: 

return - 1/self._group.n() * Y*dX\ 

- ZZ(1)/ZZ(2) * X**(self._group.n()-1) * dY\ 

- (self._group.n()-2) / (4*self._group.n()) * (Z**2+X**(self._group.n()-2)) * dZ 

 

@cached_method 

def has_reduce_hom(self): 

r""" 

Return whether the method ``reduce`` should reduce 

homogeneous elements to the corresponding space of homogeneous elements. 

 

This is mainly used by binary operations on homogeneous 

spaces which temporarily produce an element of ``self`` 

but want to consider it as a homogeneous element 

(also see ``reduce``). 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing().has_reduce_hom() 

False 

sage: ModularFormsRing(red_hom=True).has_reduce_hom() 

True 

 

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

sage: ModularForms(k=6).has_reduce_hom() 

True 

sage: ModularForms(k=6).graded_ring().has_reduce_hom() 

True 

""" 

 

return self._red_hom 

 

def is_homogeneous(self): 

r""" 

Return whether ``self`` is homogeneous component. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing().is_homogeneous() 

False 

 

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

sage: ModularForms(k=6).is_homogeneous() 

True 

""" 

 

return self._weight is not None 

 

def is_modular(self): 

r""" 

Return whether ``self`` only contains modular elements. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiWeakModularFormsRing, CuspFormsRing 

sage: QuasiWeakModularFormsRing().is_modular() 

False 

sage: CuspFormsRing(n=7).is_modular() 

True 

 

sage: from sage.modular.modform_hecketriangle.space import QuasiWeakModularForms, CuspForms 

sage: QuasiWeakModularForms(k=10).is_modular() 

False 

sage: CuspForms(n=7, k=12, base_ring=AA).is_modular() 

True 

""" 

 

return not (self.AT("quasi") <= self._analytic_type) 

 

def is_weakly_holomorphic(self): 

r""" 

Return whether ``self`` only contains weakly 

holomorphic modular elements. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, QuasiWeakModularFormsRing, CuspFormsRing 

sage: QuasiMeromorphicModularFormsRing().is_weakly_holomorphic() 

False 

sage: QuasiWeakModularFormsRing().is_weakly_holomorphic() 

True 

 

sage: from sage.modular.modform_hecketriangle.space import MeromorphicModularForms, CuspForms 

sage: MeromorphicModularForms(k=10).is_weakly_holomorphic() 

False 

sage: CuspForms(n=7, k=12, base_ring=AA).is_weakly_holomorphic() 

True 

""" 

 

return (self.AT("weak", "quasi") >= self._analytic_type) 

 

def is_holomorphic(self): 

r""" 

Return whether ``self`` only contains holomorphic 

modular elements. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiWeakModularFormsRing, QuasiModularFormsRing 

sage: QuasiWeakModularFormsRing().is_holomorphic() 

False 

sage: QuasiModularFormsRing().is_holomorphic() 

True 

 

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms, CuspForms 

sage: WeakModularForms(k=10).is_holomorphic() 

False 

sage: CuspForms(n=7, k=12, base_ring=AA).is_holomorphic() 

True 

""" 

 

return (self.AT("holo", "quasi") >= self._analytic_type) 

 

def is_cuspidal(self): 

r""" 

Return whether ``self`` only contains cuspidal elements. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing, QuasiCuspFormsRing 

sage: QuasiModularFormsRing().is_cuspidal() 

False 

sage: QuasiCuspFormsRing().is_cuspidal() 

True 

 

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

sage: ModularForms(k=12).is_cuspidal() 

False 

sage: QuasiCuspForms(k=12).is_cuspidal() 

True 

""" 

 

return (self.AT("cusp", "quasi") >= self._analytic_type) 

 

def is_zerospace(self): 

r""" 

Return whether ``self`` is the (`0`-dimensional) zero space. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing().is_zerospace() 

False 

 

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

sage: ModularForms(k=12).is_zerospace() 

False 

sage: CuspForms(k=12).reduce_type([]).is_zerospace() 

True 

""" 

 

return (self.AT(["quasi"]) >= self._analytic_type) 

 

def analytic_type(self): 

r""" 

Return the analytic type of ``self``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, QuasiWeakModularFormsRing 

sage: QuasiMeromorphicModularFormsRing().analytic_type() 

quasi meromorphic modular 

sage: QuasiWeakModularFormsRing().analytic_type() 

quasi weakly holomorphic modular 

 

sage: from sage.modular.modform_hecketriangle.space import MeromorphicModularForms, CuspForms 

sage: MeromorphicModularForms(k=10).analytic_type() 

meromorphic modular 

sage: CuspForms(n=7, k=12, base_ring=AA).analytic_type() 

cuspidal 

""" 

 

return self._analytic_type 

 

def homogeneous_part(self, k, ep): 

r""" 

Return the homogeneous component of degree (``k``, ``e``) of ``self``. 

 

INPUT: 

 

- ``k`` -- An integer. 

 

- ``ep`` -- `+1` or `-1`. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, QuasiWeakModularFormsRing 

sage: QuasiMeromorphicModularFormsRing(n=7).homogeneous_part(k=2, ep=-1) 

QuasiMeromorphicModularForms(n=7, k=2, ep=-1) over Integer Ring 

""" 

 

return self.reduce_type(degree = (k,ep)) 

 

@cached_method 

def J_inv(self): 

r""" 

Return the J-invariant (Hauptmodul) of the group of ``self``. 

It is normalized such that ``J_inv(infinity) = infinity``, 

it has real Fourier coefficients starting with ``d > 0`` and ``J_inv(i) = 1`` 

 

It lies in a (weak) extension of the graded ring of ``self``. 

In case ``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, WeakModularFormsRing, CuspFormsRing 

sage: MR = WeakModularFormsRing(n=7) 

sage: J_inv = MR.J_inv() 

sage: J_inv in MR 

True 

sage: CuspFormsRing(n=7).J_inv() == J_inv 

True 

sage: J_inv 

f_rho^7/(f_rho^7 - f_i^2) 

sage: QuasiMeromorphicModularFormsRing(n=7).J_inv() == QuasiMeromorphicModularFormsRing(n=7)(J_inv) 

True 

 

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms, CuspForms 

sage: MF = WeakModularForms(n=5, k=0) 

sage: J_inv = MF.J_inv() 

sage: J_inv in MF 

True 

sage: WeakModularFormsRing(n=5, red_hom=True).J_inv() == J_inv 

True 

sage: CuspForms(n=5, k=12).J_inv() == J_inv 

True 

sage: MF.disp_prec(3) 

sage: J_inv 

d*q^-1 + 79/200 + 42877/(640000*d)*q + 12957/(2000000*d^2)*q^2 + O(q^3) 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: MF = WeakModularForms(n=5) 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: WeakModularForms(n=5).J_inv().q_expansion(prec=5) == MFC(group=5, prec=7).J_inv_ZZ()(q/d).add_bigoh(5) 

True 

sage: WeakModularForms(n=infinity).J_inv().q_expansion(prec=5) == MFC(group=infinity, prec=7).J_inv_ZZ()(q/d).add_bigoh(5) 

True 

sage: WeakModularForms(n=5).J_inv().q_expansion(fix_d=1, prec=5) == MFC(group=5, prec=7).J_inv_ZZ().add_bigoh(5) 

True 

sage: WeakModularForms(n=infinity).J_inv().q_expansion(fix_d=1, prec=5) == MFC(group=infinity, prec=7).J_inv_ZZ().add_bigoh(5) 

True 

 

sage: WeakModularForms(n=infinity).J_inv() 

1/64*q^-1 + 3/8 + 69/16*q + 32*q^2 + 5601/32*q^3 + 768*q^4 + O(q^5) 

 

sage: WeakModularForms().J_inv() 

1/1728*q^-1 + 31/72 + 1823/16*q + 335840/27*q^2 + 16005555/32*q^3 + 11716352*q^4 + O(q^5) 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

if (self.hecke_n() == infinity): 

return self.extend_type("weak", ring=True)(x/(x-y**2)).reduce() 

else: 

return self.extend_type("weak", ring=True)(x**self._group.n()/(x**self._group.n()-y**2)).reduce() 

 

@cached_method 

def j_inv(self): 

r""" 

Return the j-invariant (Hauptmodul) of the group of ``self``. 

It is normalized such that ``j_inv(infinity) = infinity``, 

and such that it has real Fourier coefficients starting with ``1``. 

 

It lies in a (weak) extension of the graded ring of ``self``. 

In case ``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, WeakModularFormsRing, CuspFormsRing 

sage: MR = WeakModularFormsRing(n=7) 

sage: j_inv = MR.j_inv() 

sage: j_inv in MR 

True 

sage: CuspFormsRing(n=7).j_inv() == j_inv 

True 

sage: j_inv 

f_rho^7/(f_rho^7*d - f_i^2*d) 

sage: QuasiMeromorphicModularFormsRing(n=7).j_inv() == QuasiMeromorphicModularFormsRing(n=7)(j_inv) 

True 

 

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms, CuspForms 

sage: MF = WeakModularForms(n=5, k=0) 

sage: j_inv = MF.j_inv() 

sage: j_inv in MF 

True 

sage: WeakModularFormsRing(n=5, red_hom=True).j_inv() == j_inv 

True 

sage: CuspForms(n=5, k=12).j_inv() == j_inv 

True 

sage: MF.disp_prec(3) 

sage: j_inv 

q^-1 + 79/(200*d) + 42877/(640000*d^2)*q + 12957/(2000000*d^3)*q^2 + O(q^3) 

 

sage: WeakModularForms(n=infinity).j_inv() 

q^-1 + 24 + 276*q + 2048*q^2 + 11202*q^3 + 49152*q^4 + O(q^5) 

 

sage: WeakModularForms().j_inv() 

q^-1 + 744 + 196884*q + 21493760*q^2 + 864299970*q^3 + 20245856256*q^4 + O(q^5) 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

if (self.hecke_n() == infinity): 

return self.extend_type("weak", ring=True)(1/d*x/(x-y**2)).reduce() 

else: 

return self.extend_type("weak", ring=True)(1/d*x**self._group.n()/(x**self._group.n()-y**2)).reduce() 

 

@cached_method 

def f_rho(self): 

r""" 

Return a normalized modular form ``f_rho`` with exactly one simple 

zero at ``rho`` (up to the group action). 

 

It lies in a (holomorphic) extension of the graded ring of ``self``. 

In case ``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

The polynomial variable ``x`` exactly corresponds to ``f_rho``. 

 

NOTE: 

 

If ``n=infinity`` the situation is different, there we have: 

``f_rho=1`` (since that's the limit as ``n`` goes to infinity) 

and the polynomial variable ``x`` no longer refers to ``f_rho``. 

Instead it refers to ``E4`` which has exactly one simple zero 

at the cusp ``-1``. Also note that ``E4`` is the limit of 

``f_rho^(n-2)``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, ModularFormsRing, CuspFormsRing 

sage: MR = ModularFormsRing(n=7) 

sage: f_rho = MR.f_rho() 

sage: f_rho in MR 

True 

sage: CuspFormsRing(n=7).f_rho() == f_rho 

True 

sage: f_rho 

f_rho 

sage: QuasiMeromorphicModularFormsRing(n=7).f_rho() == QuasiMeromorphicModularFormsRing(n=7)(f_rho) 

True 

 

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

sage: MF = ModularForms(n=5, k=4/3) 

sage: f_rho = MF.f_rho() 

sage: f_rho in MF 

True 

sage: ModularFormsRing(n=5, red_hom=True).f_rho() == f_rho 

True 

sage: CuspForms(n=5, k=12).f_rho() == f_rho 

True 

sage: MF.disp_prec(3) 

sage: f_rho 

1 + 7/(100*d)*q + 21/(160000*d^2)*q^2 + O(q^3) 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: MF = ModularForms(n=5) 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: ModularForms(n=5).f_rho().q_expansion(prec=5) == MFC(group=5, prec=7).f_rho_ZZ()(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=infinity).f_rho().q_expansion(prec=5) == MFC(group=infinity, prec=7).f_rho_ZZ()(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=5).f_rho().q_expansion(fix_d=1, prec=5) == MFC(group=5, prec=7).f_rho_ZZ().add_bigoh(5) 

True 

sage: ModularForms(n=infinity).f_rho().q_expansion(fix_d=1, prec=5) == MFC(group=infinity, prec=7).f_rho_ZZ().add_bigoh(5) 

True 

 

sage: ModularForms(n=infinity, k=0).f_rho() == ModularForms(n=infinity, k=0)(1) 

True 

 

sage: ModularForms(k=4).f_rho() == ModularForms(k=4).E4() 

True 

sage: ModularForms(k=4).f_rho() 

1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + O(q^5) 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

if (self.hecke_n() == infinity): 

return self.extend_type("holo", ring=True)(1).reduce() 

else: 

return self.extend_type("holo", ring=True)(x).reduce() 

 

@cached_method 

def f_i(self): 

r""" 

Return a normalized modular form ``f_i`` with exactly one simple 

zero at ``i`` (up to the group action). 

 

It lies in a (holomorphic) extension of the graded ring of ``self``. 

In case ``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

The polynomial variable ``y`` exactly corresponds to ``f_i``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, ModularFormsRing, CuspFormsRing 

sage: MR = ModularFormsRing(n=7) 

sage: f_i = MR.f_i() 

sage: f_i in MR 

True 

sage: CuspFormsRing(n=7).f_i() == f_i 

True 

sage: f_i 

f_i 

sage: QuasiMeromorphicModularFormsRing(n=7).f_i() == QuasiMeromorphicModularFormsRing(n=7)(f_i) 

True 

 

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

sage: MF = ModularForms(n=5, k=10/3) 

sage: f_i = MF.f_i() 

sage: f_i in MF 

True 

sage: ModularFormsRing(n=5, red_hom=True).f_i() == f_i 

True 

sage: CuspForms(n=5, k=12).f_i() == f_i 

True 

sage: MF.disp_prec(3) 

sage: f_i 

1 - 13/(40*d)*q - 351/(64000*d^2)*q^2 + O(q^3) 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: MF = ModularForms(n=5) 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: ModularForms(n=5).f_i().q_expansion(prec=5) == MFC(group=5, prec=7).f_i_ZZ()(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=infinity).f_i().q_expansion(prec=5) == MFC(group=infinity, prec=7).f_i_ZZ()(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=5).f_i().q_expansion(fix_d=1, prec=5) == MFC(group=5, prec=7).f_i_ZZ().add_bigoh(5) 

True 

sage: ModularForms(n=infinity).f_i().q_expansion(fix_d=1, prec=5) == MFC(group=infinity, prec=7).f_i_ZZ().add_bigoh(5) 

True 

 

sage: ModularForms(n=infinity, k=2).f_i() 

1 - 24*q + 24*q^2 - 96*q^3 + 24*q^4 + O(q^5) 

 

sage: ModularForms(k=6).f_i() == ModularForms(k=4).E6() 

True 

sage: ModularForms(k=6).f_i() 

1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 + O(q^5) 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

return self.extend_type("holo", ring=True)(y).reduce() 

 

@cached_method 

def f_inf(self): 

r""" 

Return a normalized (according to its first nontrivial Fourier 

coefficient) cusp form ``f_inf`` with exactly one simple zero 

at ``infinity`` (up to the group action). 

 

It lies in a (cuspidal) extension of the graded ring of 

``self``. In case ``has_reduce_hom`` is ``True`` it is given 

as an element of the corresponding space of homogeneous elements. 

 

NOTE: 

 

If ``n=infinity`` then ``f_inf`` is no longer a cusp form 

since it doesn't vanish at the cusp ``-1``. The first 

non-trivial cusp form is given by ``E4*f_inf``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, CuspFormsRing 

sage: MR = CuspFormsRing(n=7) 

sage: f_inf = MR.f_inf() 

sage: f_inf in MR 

True 

sage: f_inf 

f_rho^7*d - f_i^2*d 

sage: QuasiMeromorphicModularFormsRing(n=7).f_inf() == QuasiMeromorphicModularFormsRing(n=7)(f_inf) 

True 

 

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

sage: MF = CuspForms(n=5, k=20/3) 

sage: f_inf = MF.f_inf() 

sage: f_inf in MF 

True 

sage: CuspFormsRing(n=5, red_hom=True).f_inf() == f_inf 

True 

sage: CuspForms(n=5, k=0).f_inf() == f_inf 

True 

sage: MF.disp_prec(3) 

sage: f_inf 

q - 9/(200*d)*q^2 + O(q^3) 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: MF = ModularForms(n=5) 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: ModularForms(n=5).f_inf().q_expansion(prec=5) == (d*MFC(group=5, prec=7).f_inf_ZZ()(q/d)).add_bigoh(5) 

True 

sage: ModularForms(n=infinity).f_inf().q_expansion(prec=5) == (d*MFC(group=infinity, prec=7).f_inf_ZZ()(q/d)).add_bigoh(5) 

True 

sage: ModularForms(n=5).f_inf().q_expansion(fix_d=1, prec=5) == MFC(group=5, prec=7).f_inf_ZZ().add_bigoh(5) 

True 

sage: ModularForms(n=infinity).f_inf().q_expansion(fix_d=1, prec=5) == MFC(group=infinity, prec=7).f_inf_ZZ().add_bigoh(5) 

True 

 

sage: ModularForms(n=infinity, k=4).f_inf().reduced_parent() 

ModularForms(n=+Infinity, k=4, ep=1) over Integer Ring 

sage: ModularForms(n=infinity, k=4).f_inf() 

q - 8*q^2 + 28*q^3 - 64*q^4 + O(q^5) 

 

sage: CuspForms(k=12).f_inf() == CuspForms(k=12).Delta() 

True 

sage: CuspForms(k=12).f_inf() 

q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5) 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

if (self.hecke_n() == infinity): 

return self.extend_type("holo", ring=True)(d*(x-y**2)).reduce() 

else: 

return self.extend_type("cusp", ring=True)(d*(x**self._group.n()-y**2)).reduce() 

 

@cached_method 

def G_inv(self): 

r""" 

If `2` divides `n`: Return the G-invariant of the group of ``self``. 

 

The G-invariant is analogous to the J-invariant but has multiplier `-1`. 

I.e. ``G_inv(-1/t) = -G_inv(t)``. It is a holomorphic square root 

of ``J_inv*(J_inv-1)`` with real Fourier coefficients. 

 

If `2` does not divide `n` the function does not exist and an 

exception is raised. 

 

The G-invariant lies in a (weak) extension of the graded ring of ``self``. 

In case ``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

NOTE: 

 

If ``n=infinity`` then ``G_inv`` is holomorphic everywhere except 

at the cusp ``-1`` where it isn't even meromorphic. Consequently 

this function raises an exception for ``n=infinity``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, WeakModularFormsRing, CuspFormsRing 

sage: MR = WeakModularFormsRing(n=8) 

sage: G_inv = MR.G_inv() 

sage: G_inv in MR 

True 

sage: CuspFormsRing(n=8).G_inv() == G_inv 

True 

sage: G_inv 

f_rho^4*f_i*d/(f_rho^8 - f_i^2) 

sage: QuasiMeromorphicModularFormsRing(n=8).G_inv() == QuasiMeromorphicModularFormsRing(n=8)(G_inv) 

True 

 

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms, CuspForms 

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

sage: G_inv = MF.G_inv() 

sage: G_inv in MF 

True 

sage: WeakModularFormsRing(n=8, red_hom=True).G_inv() == G_inv 

True 

sage: CuspForms(n=8, k=12, ep=1).G_inv() == G_inv 

True 

sage: MF.disp_prec(3) 

sage: G_inv 

d^2*q^-1 - 15*d/128 - 15139/262144*q - 11575/(1572864*d)*q^2 + O(q^3) 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: MF = WeakModularForms(n=8) 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: WeakModularForms(n=8).G_inv().q_expansion(prec=5) == (d*MFC(group=8, prec=7).G_inv_ZZ()(q/d)).add_bigoh(5) 

True 

sage: WeakModularForms(n=8).G_inv().q_expansion(fix_d=1, prec=5) == MFC(group=8, prec=7).G_inv_ZZ().add_bigoh(5) 

True 

 

sage: WeakModularForms(n=4, k=0, ep=-1).G_inv() 

1/65536*q^-1 - 3/8192 - 955/16384*q - 49/32*q^2 - 608799/32768*q^3 - 659/4*q^4 + O(q^5) 

 

As explained above, the G-invariant exists only for even `n`:: 

 

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

sage: MF = WeakModularForms(n=9) 

sage: MF.G_inv() 

Traceback (most recent call last): 

... 

ArithmeticError: G_inv doesn't exist for odd n(=9). 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

if (self.hecke_n() == infinity): 

raise ArithmeticError("G_inv doesn't exist for n={} (it is not meromorphic at -1).".format(self._group.n())) 

elif (ZZ(2).divides(self._group.n())): 

return self.extend_type("weak", ring=True)(d*y*x**(self._group.n()/ZZ(2))/(x**self._group.n()-y**2)).reduce() 

else: 

raise ArithmeticError("G_inv doesn't exist for odd n(={}).".format(self._group.n())) 

 

@cached_method 

def g_inv(self): 

r""" 

If `2` divides `n`: Return the g-invariant of the group of ``self``. 

 

The g-invariant is analogous to the j-invariant but has 

multiplier ``-1``. I.e. ``g_inv(-1/t) = -g_inv(t)``. It is a 

(normalized) holomorphic square root of ``J_inv*(J_inv-1)``, 

normalized such that its first nontrivial Fourier coefficient 

is ``1``. 

 

If `2` does not divide ``n`` the function does not exist and 

an exception is raised. 

 

The g-invariant lies in a (weak) extension of the graded ring of ``self``. 

In case ``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

NOTE: 

 

If ``n=infinity`` then ``g_inv`` is holomorphic everywhere except 

at the cusp ``-1`` where it isn't even meromorphic. Consequently 

this function raises an exception for ``n=infinity``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, WeakModularFormsRing, CuspFormsRing 

sage: MR = WeakModularFormsRing(n=8) 

sage: g_inv = MR.g_inv() 

sage: g_inv in MR 

True 

sage: CuspFormsRing(n=8).g_inv() == g_inv 

True 

sage: g_inv 

f_rho^4*f_i/(f_rho^8*d - f_i^2*d) 

sage: QuasiMeromorphicModularFormsRing(n=8).g_inv() == QuasiMeromorphicModularFormsRing(n=8)(g_inv) 

True 

 

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms, CuspForms 

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

sage: g_inv = MF.g_inv() 

sage: g_inv in MF 

True 

sage: WeakModularFormsRing(n=8, red_hom=True).g_inv() == g_inv 

True 

sage: CuspForms(n=8, k=12, ep=1).g_inv() == g_inv 

True 

sage: MF.disp_prec(3) 

sage: g_inv 

q^-1 - 15/(128*d) - 15139/(262144*d^2)*q - 11575/(1572864*d^3)*q^2 + O(q^3) 

 

sage: WeakModularForms(n=4, k=0, ep=-1).g_inv() 

q^-1 - 24 - 3820*q - 100352*q^2 - 1217598*q^3 - 10797056*q^4 + O(q^5) 

 

As explained above, the g-invariant exists only for even `n`:: 

 

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

sage: MF = WeakModularForms(n=9) 

sage: MF.g_inv() 

Traceback (most recent call last): 

... 

ArithmeticError: g_inv doesn't exist for odd n(=9). 

""" 

 

if (self.hecke_n() == infinity): 

raise ArithmeticError("g_inv doesn't exist for n={} (it is not meromorphic at -1).".format(self._group.n())) 

if (ZZ(2).divides(self._group.n())): 

(x,y,z,d) = self._pol_ring.gens() 

return self.extend_type("weak", ring=True)(1/d*y*x**(self._group.n()/ZZ(2))/(x**self._group.n()-y**2)).reduce() 

else: 

raise ArithmeticError("g_inv doesn't exist for odd n(={}).".format(self._group.n())) 

 

@cached_method 

def E4(self): 

r""" 

Return the normalized Eisenstein series of weight `4`. 

 

It lies in a (holomorphic) extension of the graded ring of ``self``. 

In case ``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

It is equal to ``f_rho^(n-2)``. 

 

NOTE: 

 

If ``n=infinity`` the situation is different, there we have: 

``f_rho=1`` (since that's the limit as ``n`` goes to infinity) 

and the polynomial variable ``x`` refers to ``E4`` instead of 

``f_rho``. In that case ``E4`` has exactly one simple zero 

at the cusp ``-1``. Also note that ``E4`` is the limit of ``f_rho^n``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, ModularFormsRing, CuspFormsRing 

sage: MR = ModularFormsRing(n=7) 

sage: E4 = MR.E4() 

sage: E4 in MR 

True 

sage: CuspFormsRing(n=7).E4() == E4 

True 

sage: E4 

f_rho^5 

sage: QuasiMeromorphicModularFormsRing(n=7).E4() == QuasiMeromorphicModularFormsRing(n=7)(E4) 

True 

 

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

sage: MF = ModularForms(n=5, k=4) 

sage: E4 = MF.E4() 

sage: E4 in MF 

True 

sage: ModularFormsRing(n=5, red_hom=True).E4() == E4 

True 

sage: CuspForms(n=5, k=12).E4() == E4 

True 

sage: MF.disp_prec(3) 

sage: E4 

1 + 21/(100*d)*q + 483/(32000*d^2)*q^2 + O(q^3) 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: MF = ModularForms(n=5) 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: ModularForms(n=5, k=4).E4().q_expansion(prec=5) == MFC(group=5, prec=7).E4_ZZ()(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=infinity, k=4).E4().q_expansion(prec=5) == MFC(group=infinity, prec=7).E4_ZZ()(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=5, k=4).E4().q_expansion(fix_d=1, prec=5) == MFC(group=5, prec=7).E4_ZZ().add_bigoh(5) 

True 

sage: ModularForms(n=infinity, k=4).E4().q_expansion(fix_d=1, prec=5) == MFC(group=infinity, prec=7).E4_ZZ().add_bigoh(5) 

True 

 

sage: ModularForms(n=infinity, k=4).E4() 

1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + O(q^5) 

 

sage: ModularForms(k=4).f_rho() == ModularForms(k=4).E4() 

True 

sage: ModularForms(k=4).E4() 

1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + O(q^5) 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

if (self.hecke_n() == infinity): 

return self.extend_type("holo", ring=True)(x).reduce() 

else: 

return self.extend_type("holo", ring=True)(x**(self._group.n()-2)).reduce() 

 

@cached_method 

def E6(self): 

r""" 

Return the normalized Eisenstein series of weight `6`. 

 

It lies in a (holomorphic) extension of the graded ring of ``self``. 

In case ``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

It is equal to ``f_rho^(n-3) * f_i``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, ModularFormsRing, CuspFormsRing 

sage: MR = ModularFormsRing(n=7) 

sage: E6 = MR.E6() 

sage: E6 in MR 

True 

sage: CuspFormsRing(n=7).E6() == E6 

True 

sage: E6 

f_rho^4*f_i 

sage: QuasiMeromorphicModularFormsRing(n=7).E6() == QuasiMeromorphicModularFormsRing(n=7)(E6) 

True 

 

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

sage: MF = ModularForms(n=5, k=6) 

sage: E6 = MF.E6() 

sage: E6 in MF 

True 

sage: ModularFormsRing(n=5, red_hom=True).E6() == E6 

True 

sage: CuspForms(n=5, k=12).E6() == E6 

True 

sage: MF.disp_prec(3) 

sage: E6 

1 - 37/(200*d)*q - 14663/(320000*d^2)*q^2 + O(q^3) 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: MF = ModularForms(n=5, k=6) 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: ModularForms(n=5, k=6).E6().q_expansion(prec=5) == MFC(group=5, prec=7).E6_ZZ()(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=infinity, k=6).E6().q_expansion(prec=5) == MFC(group=infinity, prec=7).E6_ZZ()(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=5, k=6).E6().q_expansion(fix_d=1, prec=5) == MFC(group=5, prec=7).E6_ZZ().add_bigoh(5) 

True 

sage: ModularForms(n=infinity, k=6).E6().q_expansion(fix_d=1, prec=5) == MFC(group=infinity, prec=7).E6_ZZ().add_bigoh(5) 

True 

 

sage: ModularForms(n=infinity, k=6).E6() 

1 - 8*q - 248*q^2 - 1952*q^3 - 8440*q^4 + O(q^5) 

 

sage: ModularForms(k=6).f_i() == ModularForms(k=6).E6() 

True 

sage: ModularForms(k=6).E6() 

1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 + O(q^5) 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

if (self.hecke_n() == infinity): 

return self.extend_type("holo", ring=True)(x*y).reduce() 

else: 

return self.extend_type("holo", ring=True)(x**(self._group.n()-3)*y).reduce() 

 

@cached_method 

def Delta(self): 

r""" 

Return an analog of the Delta-function. 

 

It lies in the graded ring of ``self``. In case 

``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

It is a cusp form of weight `12` and is equal to ``d*(E4^3 - 

E6^2)`` or (in terms of the generators) ``d*x^(2*n-6)*(x^n - 

y^2)``. 

 

Note that ``Delta`` is also a cusp form for ``n=infinity``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, CuspFormsRing 

sage: MR = CuspFormsRing(n=7) 

sage: Delta = MR.Delta() 

sage: Delta in MR 

True 

sage: Delta 

f_rho^15*d - f_rho^8*f_i^2*d 

sage: QuasiMeromorphicModularFormsRing(n=7).Delta() == QuasiMeromorphicModularFormsRing(n=7)(Delta) 

True 

 

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

sage: MF = CuspForms(n=5, k=12) 

sage: Delta = MF.Delta() 

sage: Delta in MF 

True 

sage: CuspFormsRing(n=5, red_hom=True).Delta() == Delta 

True 

sage: CuspForms(n=5, k=0).Delta() == Delta 

True 

sage: MF.disp_prec(3) 

sage: Delta 

q + 47/(200*d)*q^2 + O(q^3) 

 

sage: d = ModularForms(n=5).get_d() 

sage: Delta == (d*(ModularForms(n=5).E4()^3-ModularForms(n=5).E6()^2)) 

True 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: MF = CuspForms(n=5, k=12) 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: CuspForms(n=5, k=12).Delta().q_expansion(prec=5) == (d*MFC(group=5, prec=7).Delta_ZZ()(q/d)).add_bigoh(5) 

True 

sage: CuspForms(n=infinity, k=12).Delta().q_expansion(prec=5) == (d*MFC(group=infinity, prec=7).Delta_ZZ()(q/d)).add_bigoh(5) 

True 

sage: CuspForms(n=5, k=12).Delta().q_expansion(fix_d=1, prec=5) == MFC(group=5, prec=7).Delta_ZZ().add_bigoh(5) 

True 

sage: CuspForms(n=infinity, k=12).Delta().q_expansion(fix_d=1, prec=5) == MFC(group=infinity, prec=7).Delta_ZZ().add_bigoh(5) 

True 

 

sage: CuspForms(n=infinity, k=12).Delta() 

q + 24*q^2 + 252*q^3 + 1472*q^4 + O(q^5) 

 

sage: CuspForms(k=12).f_inf() == CuspForms(k=12).Delta() 

True 

sage: CuspForms(k=12).Delta() 

q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5) 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

if (self.hecke_n() == infinity): 

return self.extend_type("cusp", ring=True)(d*x**2*(x-y**2)).reduce() 

else: 

return self.extend_type("cusp", ring=True)(d*x**(2*self._group.n()-6)*(x**self._group.n()-y**2)).reduce() 

 

@cached_method 

def E2(self): 

r""" 

Return the normalized quasi holomorphic Eisenstein series of weight `2`. 

 

It lies in a (quasi holomorphic) extension of the graded ring of ``self``. 

In case ``has_reduce_hom`` is ``True`` it is given as an element of 

the corresponding space of homogeneous elements. 

 

It is in particular also a generator of the graded ring of 

``self`` and the polynomial variable ``z`` exactly corresponds to ``E2``. 

 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing, QuasiModularFormsRing, CuspFormsRing 

sage: MR = QuasiModularFormsRing(n=7) 

sage: E2 = MR.E2() 

sage: E2 in MR 

True 

sage: CuspFormsRing(n=7).E2() == E2 

True 

sage: E2 

E2 

sage: QuasiMeromorphicModularFormsRing(n=7).E2() == QuasiMeromorphicModularFormsRing(n=7)(E2) 

True 

 

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

sage: MF = QuasiModularForms(n=5, k=2) 

sage: E2 = MF.E2() 

sage: E2 in MF 

True 

sage: QuasiModularFormsRing(n=5, red_hom=True).E2() == E2 

True 

sage: CuspForms(n=5, k=12, ep=1).E2() == E2 

True 

sage: MF.disp_prec(3) 

sage: E2 

1 - 9/(200*d)*q - 369/(320000*d^2)*q^2 + O(q^3) 

 

sage: f_inf = MF.f_inf() 

sage: E2 == f_inf.derivative() / f_inf 

True 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: MF = QuasiModularForms(n=5, k=2) 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: QuasiModularForms(n=5, k=2).E2().q_expansion(prec=5) == MFC(group=5, prec=7).E2_ZZ()(q/d).add_bigoh(5) 

True 

sage: QuasiModularForms(n=infinity, k=2).E2().q_expansion(prec=5) == MFC(group=infinity, prec=7).E2_ZZ()(q/d).add_bigoh(5) 

True 

sage: QuasiModularForms(n=5, k=2).E2().q_expansion(fix_d=1, prec=5) == MFC(group=5, prec=7).E2_ZZ().add_bigoh(5) 

True 

sage: QuasiModularForms(n=infinity, k=2).E2().q_expansion(fix_d=1, prec=5) == MFC(group=infinity, prec=7).E2_ZZ().add_bigoh(5) 

True 

 

sage: QuasiModularForms(n=infinity, k=2).E2() 

1 - 8*q - 8*q^2 - 32*q^3 - 40*q^4 + O(q^5) 

 

sage: QuasiModularForms(k=2).E2() 

1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 + O(q^5) 

""" 

 

(x,y,z,d) = self._pol_ring.gens() 

 

return self.extend_type(["holo", "quasi"], ring=True)(z).reduce() 

 

@cached_method 

def EisensteinSeries(self, k=None): 

r""" 

Return the normalized Eisenstein series of weight ``k``. 

 

Only arithmetic groups or trivial weights (with corresponding 

one dimensional spaces) are supported. 

 

INPUT: 

 

- ``k`` -- A non-negative even integer, namely the weight. 

 

If ``k=None`` (default) then the weight of ``self`` 

is choosen if ``self`` is homogeneous and the 

weight is possible, otherwise ``k=0`` is set. 

 

OUTPUT: 

 

A modular form element lying in a (holomorphic) extension of 

the graded ring of ``self``. In case ``has_reduce_hom`` is 

``True`` it is given as an element of the corresponding 

space of homogeneous elements. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing, CuspFormsRing 

sage: MR = ModularFormsRing() 

sage: MR.EisensteinSeries() == MR.one() 

True 

sage: E8 = MR.EisensteinSeries(k=8) 

sage: E8 in MR 

True 

sage: E8 

f_rho^2 

 

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

sage: MF = ModularForms(n=4, k=12) 

sage: E12 = MF.EisensteinSeries() 

sage: E12 in MF 

True 

sage: CuspFormsRing(n=4, red_hom=True).EisensteinSeries(k=12).parent() 

ModularForms(n=4, k=12, ep=1) over Integer Ring 

sage: MF.disp_prec(4) 

sage: E12 

1 + 1008/691*q + 2129904/691*q^2 + 178565184/691*q^3 + O(q^4) 

 

sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor as MFC 

sage: d = MF.get_d() 

sage: q = MF.get_q() 

sage: ModularForms(n=3, k=2).EisensteinSeries().q_expansion(prec=5) == MFC(group=3, prec=7).EisensteinSeries_ZZ(k=2)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=3, k=4).EisensteinSeries().q_expansion(prec=5) == MFC(group=3, prec=7).EisensteinSeries_ZZ(k=4)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=3, k=6).EisensteinSeries().q_expansion(prec=5) == MFC(group=3, prec=7).EisensteinSeries_ZZ(k=6)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=3, k=8).EisensteinSeries().q_expansion(prec=5) == MFC(group=3, prec=7).EisensteinSeries_ZZ(k=8)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=4, k=2).EisensteinSeries().q_expansion(prec=5) == MFC(group=4, prec=7).EisensteinSeries_ZZ(k=2)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=4, k=4).EisensteinSeries().q_expansion(prec=5) == MFC(group=4, prec=7).EisensteinSeries_ZZ(k=4)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=4, k=6).EisensteinSeries().q_expansion(prec=5) == MFC(group=4, prec=7).EisensteinSeries_ZZ(k=6)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=4, k=8).EisensteinSeries().q_expansion(prec=5) == MFC(group=4, prec=7).EisensteinSeries_ZZ(k=8)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=6, k=2, ep=-1).EisensteinSeries().q_expansion(prec=5) == MFC(group=6, prec=7).EisensteinSeries_ZZ(k=2)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=6, k=4).EisensteinSeries().q_expansion(prec=5) == MFC(group=6, prec=7).EisensteinSeries_ZZ(k=4)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=6, k=6, ep=-1).EisensteinSeries().q_expansion(prec=5) == MFC(group=6, prec=7).EisensteinSeries_ZZ(k=6)(q/d).add_bigoh(5) 

True 

sage: ModularForms(n=6, k=8).EisensteinSeries().q_expansion(prec=5) == MFC(group=6, prec=7).EisensteinSeries_ZZ(k=8)(q/d).add_bigoh(5) 

True 

 

sage: ModularForms(n=3, k=12).EisensteinSeries() 

1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + O(q^5) 

sage: ModularForms(n=4, k=12).EisensteinSeries() 

1 + 1008/691*q + 2129904/691*q^2 + 178565184/691*q^3 + O(q^4) 

sage: ModularForms(n=6, k=12).EisensteinSeries() 

1 + 6552/50443*q + 13425048/50443*q^2 + 1165450104/50443*q^3 + 27494504856/50443*q^4 + O(q^5) 

sage: ModularForms(n=3, k=20).EisensteinSeries() 

1 + 13200/174611*q + 6920614800/174611*q^2 + 15341851377600/174611*q^3 + 3628395292275600/174611*q^4 + O(q^5) 

sage: ModularForms(n=4).EisensteinSeries(k=8) 

1 + 480/17*q + 69600/17*q^2 + 1050240/17*q^3 + 8916960/17*q^4 + O(q^5) 

sage: ModularForms(n=6).EisensteinSeries(k=20) 

1 + 264/206215591*q + 138412296/206215591*q^2 + 306852616488/206215591*q^3 + 72567905845512/206215591*q^4 + O(q^5) 

""" 

 

n = self.hecke_n() 

 

# For now we completely disable Eisenstein series for n == infinity, 

# but leave some related basic variables intact. 

if n == infinity: 

raise NotImplementedError("In the case n=infinity, the Eisenstein series is not unique and more parameters are required.") 

 

if (k is None): 

try: 

if not self.is_homogeneous(): 

raise TypeError(None) 

k = self.weight() 

if k < 0: 

raise TypeError(None) 

k = 2*ZZ(k/2) 

#if self.ep() != ZZ(-1)**ZZ(k/2): 

# raise TypeError 

except TypeError: 

k = ZZ(0) 

 

try: 

if k < 0: 

raise TypeError(None) 

k = 2*ZZ(k/2) 

except TypeError: 

raise TypeError("k={} must be a non-negative even integer!".format(k)) 

 

# The case n=infinity is special (there are 2 cusps) 

# Until we/I get confirmation what is what sort of Eisenstein series 

# this case is excluded... 

if n == infinity: 

# We set the weight zero Eisenstein series to 1 

pass 

elif k == 0: 

return self.one() 

elif k == 2: 

# This is a bit problematic, e.g. for n=infinity there is a 

# classical Eisenstein series of weight 2 

return self.E2() 

elif k == 4: 

return self.E4() 

elif k == 6: 

return self.E6() 

 

# Basic variables 

ep = (-ZZ(1))**(k/2) 

extended_self = self.extend_type(["holo"], ring=True) 

# reduced_self is a classical ModularForms space 

reduced_self = extended_self.reduce_type(["holo"], degree = (QQ(k), ep)) 

 

if (n == infinity): 

l2 = ZZ(0) 

l1 = ZZ((k-(1-ep)) / ZZ(4)) 

else: 

num = ZZ((k-(1-ep)*n/(n-2)) * (n-2) / ZZ(4)) 

l2 = num % n 

l1 = ((num-l2)/n).numerator() 

 

# If the space is one dimensional we return the normalized generator 

if l1 == 0: 

return extended_self(reduced_self.F_basis(0)).reduce() 

 

# The non-arithmetic remaining cases (incomplete, very hard in general) 

# TODO: the n = infinity case(s) (doable) 

# TODO: the n = 5 case (hard) 

if (not self.group().is_arithmetic() or n == infinity): 

raise NotImplementedError("Eisenstein series are only supported in the finite arithmetic cases") 

 

# The arithmetic cases 

prec = reduced_self._l1 + 1 

MFC = MFSeriesConstructor(group=self.group(), prec=prec) 

d = self.get_d() 

q = self.get_q() 

q_series = MFC.EisensteinSeries_ZZ(k=k)(q/d) 

 

return extended_self(reduced_self.construct_form(q_series, check=False)).reduce()