Modular forms for Hecke triangle groups

AUTHORS:

• Jonas Jermann (2013): initial version

class sage.modular.modform_hecketriangle.abstract_space.FormsSpace_abstract(group, base_ring, k, ep, n)

Bases: sage.modular.modform_hecketriangle.abstract_ring.FormsRing_abstract

Abstract (Hecke) forms space.

This should never be called directly. Instead one should instantiate one of the derived classes of this class.

Element

alias of sage.modular.modform_hecketriangle.element.FormsElement

F_basis(m, order_1=0)

Returns a weakly holomorphic element of self (extended if necessarily) determined by the property that the Fourier expansion is of the form is of the form q^m + O(q^(order_inf + 1)), where order_inf = self._l1 - order_1.

In particular for all m <= order_inf these elements form a basis of the space of weakly holomorphic modular forms of the corresponding degree in case n!=infinity.

If n=infinity a non-trivial order of -1 can be specified through the parameter order_1 (default: 0). Otherwise it is ignored.

INPUT:

• m – An integer m <= self._l1.

• order_1 – The order at -1 of F_simple (default: 0).

  This parameter is ignored if n != infinity.

OUTPUT:

The corresponding element in (possibly an extension of) self. Note that the order at -1 of the resulting element may be bigger than order_1 (rare).

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms, CuspForms
sage: MF = WeakModularForms(n=5, k=62/3, ep=-1)
sage: MF.disp_prec(MF._l1+2)
sage: MF.weight_parameters()
(2, 3)

sage: MF.F_basis(2)
q^2 - 41/(200*d)*q^3 + O(q^4)
sage: MF.F_basis(1)
q - 13071/(640000*d^2)*q^3 + O(q^4)
sage: MF.F_basis(0)
1 - 277043/(192000000*d^3)*q^3 + O(q^4)
sage: MF.F_basis(-2)
q^-2 - 162727620113/(40960000000000000*d^5)*q^3 + O(q^4)
sage: MF.F_basis(-2).parent() == MF
True

sage: MF = CuspForms(n=4, k=-2, ep=1)
sage: MF.weight_parameters()
(-1, 3)

sage: MF.F_basis(-1).parent()
WeakModularForms(n=4, k=-2, ep=1) over Integer Ring
sage: MF.F_basis(-1).parent().disp_prec(MF._l1+2)
sage: MF.F_basis(-1)
q^-1 + 80 + O(q)
sage: MF.F_basis(-2)
q^-2 + 400 + O(q)

sage: MF = WeakModularForms(n=infinity, k=14, ep=-1)
sage: MF.F_basis(3)
q^3 - 48*q^4 + O(q^5)
sage: MF.F_basis(2)
q^2 - 1152*q^4 + O(q^5)
sage: MF.F_basis(1)
q - 18496*q^4 + O(q^5)
sage: MF.F_basis(0)
1 - 224280*q^4 + O(q^5)
sage: MF.F_basis(-1)
q^-1 - 2198304*q^4 + O(q^5)

sage: MF.F_basis(3, order_1=-1)
q^3 + O(q^5)
sage: MF.F_basis(1, order_1=2)
q - 300*q^3 - 4096*q^4 + O(q^5)
sage: MF.F_basis(0, order_1=2)
1 - 24*q^2 - 2048*q^3 - 98328*q^4 + O(q^5)
sage: MF.F_basis(-1, order_1=2)
q^-1 - 18150*q^3 - 1327104*q^4 + O(q^5)

F_basis_pol(m, order_1=0)

Returns a polynomial corresponding to the basis element of the corresponding space of weakly holomorphic forms of the same degree as self. The basis element is determined by the property that the Fourier expansion is of the form q^m + O(q^(order_inf + 1)), where order_inf = self._l1 - order_1.

If n=infinity a non-trivial order of -1 can be specified through the parameter order_1 (default: 0). Otherwise it is ignored.

INPUT:

• m – An integer m <= self._l1.

• order_1 – The order at -1 of F_simple (default: 0).

  This parameter is ignored if n != infinity.

OUTPUT:

A polynomial in x,y,z,d, corresponding to f_rho, f_i, E2 and the (possibly) transcendental parameter d.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms
sage: MF = WeakModularForms(n=5, k=62/3, ep=-1)
sage: MF.weight_parameters()
(2, 3)

sage: MF.F_basis_pol(2)
x^13*y*d^2 - 2*x^8*y^3*d^2 + x^3*y^5*d^2
sage: MF.F_basis_pol(1)
(-81*x^13*y*d + 62*x^8*y^3*d + 19*x^3*y^5*d)/(-100)
sage: MF.F_basis_pol(0)
(141913*x^13*y + 168974*x^8*y^3 + 9113*x^3*y^5)/320000

sage: MF(MF.F_basis_pol(2)).q_expansion(prec=MF._l1+2)
q^2 - 41/(200*d)*q^3 + O(q^4)
sage: MF(MF.F_basis_pol(1)).q_expansion(prec=MF._l1+1)
q + O(q^3)
sage: MF(MF.F_basis_pol(0)).q_expansion(prec=MF._l1+1)
1 + O(q^3)
sage: MF(MF.F_basis_pol(-2)).q_expansion(prec=MF._l1+1)
q^-2 + O(q^3)
sage: MF(MF.F_basis_pol(-2)).parent()
WeakModularForms(n=5, k=62/3, ep=-1) over Integer Ring

sage: MF = WeakModularForms(n=4, k=-2, ep=1)
sage: MF.weight_parameters()
(-1, 3)

sage: MF.F_basis_pol(-1)
x^3/(x^4*d - y^2*d)
sage: MF.F_basis_pol(-2)
(9*x^7 + 23*x^3*y^2)/(32*x^8*d^2 - 64*x^4*y^2*d^2 + 32*y^4*d^2)

sage: MF(MF.F_basis_pol(-1)).q_expansion(prec=MF._l1+2)
q^-1 + 5/(16*d) + O(q)
sage: MF(MF.F_basis_pol(-2)).q_expansion(prec=MF._l1+2)
q^-2 + 25/(4096*d^2) + O(q)

sage: MF = WeakModularForms(n=infinity, k=14, ep=-1)
sage: MF.F_basis_pol(3)
-y^7*d^3 + 3*x*y^5*d^3 - 3*x^2*y^3*d^3 + x^3*y*d^3
sage: MF.F_basis_pol(2)
(3*y^7*d^2 - 17*x*y^5*d^2 + 25*x^2*y^3*d^2 - 11*x^3*y*d^2)/(-8)
sage: MF.F_basis_pol(1)
(-75*y^7*d + 225*x*y^5*d - 1249*x^2*y^3*d + 1099*x^3*y*d)/1024
sage: MF.F_basis_pol(0)
(41*y^7 - 147*x*y^5 - 1365*x^2*y^3 - 2625*x^3*y)/(-4096)
sage: MF.F_basis_pol(-1)
(-9075*y^9 + 36300*x*y^7 - 718002*x^2*y^5 - 4928052*x^3*y^3 - 2769779*x^4*y)/(8388608*y^2*d - 8388608*x*d)

sage: MF.F_basis_pol(3, order_1=-1)
(-3*y^9*d^3 + 16*x*y^7*d^3 - 30*x^2*y^5*d^3 + 24*x^3*y^3*d^3 - 7*x^4*y*d^3)/(-4*x)
sage: MF.F_basis_pol(1, order_1=2)
-x^2*y^3*d + x^3*y*d
sage: MF.F_basis_pol(0, order_1=2)
(-3*x^2*y^3 - 5*x^3*y)/(-8)
sage: MF.F_basis_pol(-1, order_1=2)
(-81*x^2*y^5 - 606*x^3*y^3 - 337*x^4*y)/(1024*y^2*d - 1024*x*d)

F_simple(order_1=0)

Return a (the most) simple normalized element of self corresponding to the weight parameters l1=self._l1 and l2=self._l2. If the element does not lie in self the type of its parent is extended accordingly.

The main part of the element is given by the (l1 - order_1)-th power of f_inf, up to a small holomorphic correction factor.

INPUT:

• order_1 – An integer (default: 0) denoting the desired order at

  -1 in the case n = infinity. If n != infinity the parameter is ignored.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms
sage: MF = WeakModularForms(n=18, k=-7, ep=-1)
sage: MF.disp_prec(1)
sage: MF.F_simple()
q^-3 + 16/(81*d)*q^-2 - 4775/(104976*d^2)*q^-1 - 14300/(531441*d^3) + O(q)
sage: MF.F_simple() == MF.f_inf()^MF._l1 * MF.f_rho()^MF._l2 * MF.f_i()
True

sage: from sage.modular.modform_hecketriangle.space import CuspForms, ModularForms
sage: MF = CuspForms(n=5, k=2, ep=-1)
sage: MF._l1
-1
sage: MF.F_simple().parent()
WeakModularForms(n=5, k=2, ep=-1) over Integer Ring

sage: MF = ModularForms(n=infinity, k=8, ep=1)
sage: MF.F_simple().reduced_parent()
ModularForms(n=+Infinity, k=8, ep=1) over Integer Ring
sage: MF.F_simple()
q^2 - 16*q^3 + 120*q^4 + O(q^5)
sage: MF.F_simple(order_1=2)
1 + 32*q + 480*q^2 + 4480*q^3 + 29152*q^4 + O(q^5)

Faber_pol(m, order_1=0, fix_d=False, d_num_prec=None)

Return the m’th Faber polynomial of self.

Namely a polynomial P(q) such that P(J_inv)*F_simple(order_1) has a Fourier expansion of the form q^m + O(q^(order_inf + 1)). where order_inf = self._l1 - order_1 and d^(order_inf - m)*P(q) is a monic polynomial of degree order_inf - m.

If n=infinity a non-trivial order of -1 can be specified through the parameter order_1 (default: 0). Otherwise it is ignored.

The Faber polynomials are e.g. used to construct a basis of weakly holomorphic forms and to recover such forms from their initial Fourier coefficients.

INPUT:

• m – An integer m <= order_inf = self._l1 - order_1.

• order_1 – The order at -1 of F_simple (default: 0).

  This parameter is ignored if n != infinity.

• 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 corresponding Faber polynomial P(q).

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms
sage: MF = WeakModularForms(n=5, k=62/3, ep=-1)
sage: MF.weight_parameters()
(2, 3)

sage: MF.Faber_pol(2)
1
sage: MF.Faber_pol(1)
1/d*q - 19/(100*d)
sage: MF.Faber_pol(0)
1/d^2*q^2 - 117/(200*d^2)*q + 9113/(320000*d^2)
sage: MF.Faber_pol(-2)
1/d^4*q^4 - 11/(8*d^4)*q^3 + 41013/(80000*d^4)*q^2 - 2251291/(48000000*d^4)*q + 1974089431/(4915200000000*d^4)
sage: (MF.Faber_pol(2)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1+2)
q^2 - 41/(200*d)*q^3 + O(q^4)
sage: (MF.Faber_pol(1)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1+1)
q + O(q^3)
sage: (MF.Faber_pol(0)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1+1)
1 + O(q^3)
sage: (MF.Faber_pol(-2)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1+1)
q^-2 + O(q^3)

sage: MF.Faber_pol(2, fix_d=1)
1
sage: MF.Faber_pol(1, fix_d=1)
q - 19/100
sage: MF.Faber_pol(-2, fix_d=1)
q^4 - 11/8*q^3 + 41013/80000*q^2 - 2251291/48000000*q + 1974089431/4915200000000
sage: (MF.Faber_pol(2, fix_d=1)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1+2, fix_d=1)
q^2 - 41/200*q^3 + O(q^4)
sage: (MF.Faber_pol(-2)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1+1, fix_d=1)
q^-2 + O(q^3)

sage: MF = WeakModularForms(n=4, k=-2, ep=1)
sage: MF.weight_parameters()
(-1, 3)

sage: MF.Faber_pol(-1)
1
sage: MF.Faber_pol(-2, fix_d=True)
256*q - 184
sage: MF.Faber_pol(-3, fix_d=True)
65536*q^2 - 73728*q + 14364
sage: (MF.Faber_pol(-1, fix_d=True)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1+2, fix_d=True)
q^-1 + 80 + O(q)
sage: (MF.Faber_pol(-2, fix_d=True)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1+2, fix_d=True)
q^-2 + 400 + O(q)
sage: (MF.Faber_pol(-3)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1+2, fix_d=True)
q^-3 + 2240 + O(q)

sage: MF = WeakModularForms(n=infinity, k=14, ep=-1)
sage: MF.Faber_pol(3)
1
sage: MF.Faber_pol(2)
1/d*q + 3/(8*d)
sage: MF.Faber_pol(1)
1/d^2*q^2 + 75/(1024*d^2)
sage: MF.Faber_pol(0)
1/d^3*q^3 - 3/(8*d^3)*q^2 + 3/(512*d^3)*q + 41/(4096*d^3)
sage: MF.Faber_pol(-1)
1/d^4*q^4 - 3/(4*d^4)*q^3 + 81/(1024*d^4)*q^2 + 9075/(8388608*d^4)
sage: (MF.Faber_pol(-1)(MF.J_inv())*MF.F_simple()).q_expansion(prec=MF._l1 + 1)
q^-1 + O(q^4)

sage: MF.Faber_pol(3, order_1=-1)
1/d*q + 3/(4*d)
sage: MF.Faber_pol(1, order_1=2)
1
sage: MF.Faber_pol(0, order_1=2)
1/d*q - 3/(8*d)
sage: MF.Faber_pol(-1, order_1=2)
1/d^2*q^2 - 3/(4*d^2)*q + 81/(1024*d^2)
sage: (MF.Faber_pol(-1, order_1=2)(MF.J_inv())*MF.F_simple(order_1=2)).q_expansion(prec=MF._l1 + 1)
q^-1 - 9075/(8388608*d^4)*q^3 + O(q^4)

FormsElement

alias of sage.modular.modform_hecketriangle.element.FormsElement

ambient_coordinate_vector(v)

Return the coordinate vector of the element v in self.module() with respect to the basis from self.ambient_space.

NOTE:

Elements use this method (from their parent) to calculate their coordinates.

INPUT:

• v – An element of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=4, k=24, ep=-1)
sage: MF.ambient_coordinate_vector(MF.gen(0)).parent()
Vector space of dimension 3 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.ambient_coordinate_vector(MF.gen(0))
(1, 0, 0)
sage: subspace = MF.subspace([MF.gen(0), MF.gen(2)])
sage: subspace.ambient_coordinate_vector(subspace.gen(0)).parent()
Vector space of degree 3 and dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
Basis matrix:
[1 0 0]
[0 0 1]
sage: subspace.ambient_coordinate_vector(subspace.gen(0))
(1, 0, 0)

ambient_module()

Return the module associated to the ambient space of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(k=12)
sage: MF.ambient_module()
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.ambient_module() == MF.module()
True
sage: subspace = MF.subspace([MF.gen(0)])
sage: subspace.ambient_module() == MF.module()
True

ambient_space()

Return the ambient space of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(k=12)
sage: MF.ambient_space()
ModularForms(n=3, k=12, ep=1) over Integer Ring
sage: MF.ambient_space() == MF
True
sage: subspace = MF.subspace([MF.gen(0)])
sage: subspace
Subspace of dimension 1 of ModularForms(n=3, k=12, ep=1) over Integer Ring
sage: subspace.ambient_space() == MF
True

aut_factor(gamma, t)

The automorphy factor of self.

INPUT:

• gamma – An element of the group of self.
• t – An element of the upper half plane.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=8, k=4, ep=1)
sage: full_factor = lambda mat, t: (mat[1][0]*t+mat[1][1])**4
sage: T = MF.group().T()
sage: S = MF.group().S()
sage: i = AlgebraicField()(i)
sage: z = 1 + i/2

sage: MF.aut_factor(S, z)
3/2*I - 7/16
sage: MF.aut_factor(-T^(-2), z)
1
sage: MF.aut_factor(MF.group().V(6), z)
173.2640595631...? + 343.8133289126...?*I
sage: MF.aut_factor(S, z) == full_factor(S, z)
True
sage: MF.aut_factor(T, z) == full_factor(T, z)
True
sage: MF.aut_factor(MF.group().V(6), z) == full_factor(MF.group().V(6), z)
True

sage: MF = ModularForms(n=7, k=14/5, ep=-1)
sage: T = MF.group().T()
sage: S = MF.group().S()

sage: MF.aut_factor(S, z)
1.3655215324256...? + 0.056805991182877...?*I
sage: MF.aut_factor(-T^(-2), z)
1
sage: MF.aut_factor(S, z) == MF.ep() * (z/i)^MF.weight()
True
sage: MF.aut_factor(MF.group().V(6), z)
13.23058830577...? + 15.71786610686...?*I

change_ring(new_base_ring)

Return the same space as self but over a new base ring new_base_ring.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import CuspForms
sage: CuspForms(n=5, k=24).change_ring(CC)
CuspForms(n=5, k=24, ep=1) over Complex Field with 53 bits of precision

construct_form(laurent_series, order_1=0, check=True, rationalize=False)

Tries to construct an element of self with the given Fourier expansion. The assumption is made that the specified Fourier expansion corresponds to a weakly holomorphic modular form.

If the precision is too low to determine the element an exception is raised.

INPUT:

• laurent_series – A Laurent or Power series.

• order_1 – A lower bound for the order at -1 of the form (default: 0).

  If n!=infinity this parameter is ignored.

• check – If True (default) then the series expansion of the constructed

  form is compared against the given series.

• rationalize – If True (default: False) then the series is

  \(rationalized\) beforehand. Note that in non-exact or non-arithmetic cases this is experimental and extremely unreliable!

OUTPUT:

If possible: An element of self with the same initial Fourier expansion as laurent_series.

Note: For modular spaces it is also possible to call self(laurent_series) instead.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import CuspForms
sage: Delta = CuspForms(k=12).Delta()
sage: qexp = Delta.q_expansion(prec=2)
sage: qexp.parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: qexp
q + O(q^2)
sage: CuspForms(k=12).construct_form(qexp) == Delta
True

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms
sage: J_inv = WeakModularForms(n=7).J_inv()
sage: qexp2 = J_inv.q_expansion(prec=1)
sage: qexp2.parent()
Laurent Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: qexp2
d*q^-1 + 151/392 + O(q)
sage: WeakModularForms(n=7).construct_form(qexp2) == J_inv
True

sage: MF = WeakModularForms(n=5, k=62/3, ep=-1)
sage: MF.default_prec(MF._l1+1)
sage: d = MF.get_d()
sage: MF.weight_parameters()
(2, 3)
sage: el2 = d*MF.F_basis(2) + 2*MF.F_basis(1) + MF.F_basis(-2)
sage: qexp2 = el2.q_expansion()
sage: qexp2.parent()
Laurent Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: qexp2
q^-2 + 2*q + d*q^2 + O(q^3)
sage: WeakModularForms(n=5, k=62/3, ep=-1).construct_form(qexp2) == el2
True

sage: MF = WeakModularForms(n=infinity, k=-2, ep=-1)
sage: el3 = MF.f_i()/MF.f_inf() + MF.f_i()*MF.f_inf()/MF.E4()^2
sage: MF.quasi_part_dimension(min_exp=-1, order_1=-2)
3
sage: prec = MF._l1 + 3
sage: qexp3 = el3.q_expansion(prec)
sage: qexp3
q^-1 - 1/(4*d) + ((1024*d^2 - 33)/(1024*d^2))*q + O(q^2)
sage: MF.construct_form(qexp3, order_1=-2) == el3
True
sage: MF.construct_form(el3.q_expansion(prec + 1), order_1=-3) == el3
True

sage: WF = WeakModularForms(n=14)
sage: qexp = WF.J_inv().q_expansion_fixed_d(d_num_prec=1000)
sage: qexp.parent()
Laurent Series Ring in q over Real Field with 1000 bits of precision
sage: WF.construct_form(qexp, rationalize=True) == WF.J_inv()
doctest:...: UserWarning: Using an experimental rationalization of coefficients, please check the result for correctness!
True

construct_quasi_form(laurent_series, order_1=0, check=True, rationalize=False)

Try to construct an element of self with the given Fourier expansion. The assumption is made that the specified Fourier expansion corresponds to a weakly holomorphic quasi modular form.

If the precision is too low to determine the element an exception is raised.

INPUT:

• laurent_series – A Laurent or Power series.

• order_1 – A lower bound for the order at -1 for all quasi parts of the

  form (default: 0). If n!=infinity this parameter is ignored.

• check – If True (default) then the series expansion of the constructed

  form is compared against the given (rationalized) series.

• rationalize – If True (default: False) then the series is

  \(rationalized\) beforehand. Note that in non-exact or non-arithmetic cases this is experimental and extremely unreliable!

OUTPUT:

If possible: An element of self with the same initial Fourier expansion as laurent_series.

Note: For non modular spaces it is also possible to call self(laurent_series) instead. Also note that this function works much faster if a corresponding (cached) q_basis is available.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiWeakModularForms, ModularForms, QuasiModularForms, QuasiCuspForms
sage: QF = QuasiWeakModularForms(n=8, k=10/3, ep=-1)
sage: el = QF.quasi_part_gens(min_exp=-1)[4]
sage: prec = QF.required_laurent_prec(min_exp=-1)
sage: prec
5
sage: qexp = el.q_expansion(prec=prec)
sage: qexp
q^-1 - 19/(64*d) - 7497/(262144*d^2)*q + 15889/(8388608*d^3)*q^2 + 543834047/(1649267441664*d^4)*q^3 + 711869853/(43980465111040*d^5)*q^4 + O(q^5)
sage: qexp.parent()
Laurent Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: constructed_el = QF.construct_quasi_form(qexp)
sage: constructed_el.parent()
QuasiWeakModularForms(n=8, k=10/3, ep=-1) over Integer Ring
sage: el == constructed_el
True

If a q_basis is available the construction uses a different algorithm which we also check::

sage: basis = QF.q_basis(min_exp=-1)
sage: QF(qexp) == constructed_el
True

sage: MF = ModularForms(k=36)
sage: el2 = MF.quasi_part_gens(min_exp=2)[1]
sage: prec = MF.required_laurent_prec(min_exp=2)
sage: prec
4
sage: qexp2 = el2.q_expansion(prec=prec + 1)
sage: qexp2
q^3 - 1/(24*d)*q^4 + O(q^5)
sage: qexp2.parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: constructed_el2 = MF.construct_quasi_form(qexp2)
sage: constructed_el2.parent()
ModularForms(n=3, k=36, ep=1) over Integer Ring
sage: el2 == constructed_el2
True

sage: QF = QuasiModularForms(k=2)
sage: q = QF.get_q()
sage: qexp3 = 1 + O(q)
sage: QF(qexp3)
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 + O(q^5)
sage: QF(qexp3) == QF.E2()
True

sage: QF = QuasiWeakModularForms(n=infinity, k=2, ep=-1)
sage: el4 = QF.f_i() + QF.f_i()^3/QF.E4()
sage: prec = QF.required_laurent_prec(order_1=-1)
sage: qexp4 = el4.q_expansion(prec=prec)
sage: qexp4
2 - 7/(4*d)*q + 195/(256*d^2)*q^2 - 903/(4096*d^3)*q^3 + 41987/(1048576*d^4)*q^4 - 181269/(33554432*d^5)*q^5 + O(q^6)
sage: QF.construct_quasi_form(qexp4, check=False) == el4
False
sage: QF.construct_quasi_form(qexp4, order_1=-1) == el4
True

sage: QF = QuasiCuspForms(n=8, k=22/3, ep=-1)
sage: el = QF(QF.f_inf()*QF.E2())
sage: qexp = el.q_expansion_fixed_d(d_num_prec=1000)
sage: qexp.parent()
Power Series Ring in q over Real Field with 1000 bits of precision
sage: QF.construct_quasi_form(qexp, rationalize=True) == el
True

construction()

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

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: QuasiModularForms(n=4, k=2, ep=1, base_ring=CC).construction()
(QuasiModularFormsFunctor(n=4, k=2, ep=1),
 BaseFacade(Complex Field with 53 bits of precision))

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF=ModularForms(k=12)
sage: MF.subspace([MF.gen(1)]).construction()
(FormsSubSpaceFunctor with 1 generator for the ModularFormsFunctor(n=3, k=12, ep=1), BaseFacade(Integer Ring))

contains_coeff_ring()

Return whether self contains its coefficient ring.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: QuasiModularForms(k=0, ep=1, n=8).contains_coeff_ring()
True
sage: QuasiModularForms(k=0, ep=-1, n=8).contains_coeff_ring()
False

coordinate_vector(v)

This method should be overloaded by subclasses.

Return the coordinate vector of the element v with respect to self.gens().

NOTE:

Elements use this method (from their parent) to calculate their coordinates.

INPUT:

• v – An element of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=4, k=24, ep=-1)
sage: MF.coordinate_vector(MF.gen(0)).parent() # defined in space.py
Vector space of dimension 3 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF.coordinate_vector(MF.gen(0))          # defined in space.py
(1, 0, 0)
sage: subspace = MF.subspace([MF.gen(0), MF.gen(2)])
sage: subspace.coordinate_vector(subspace.gen(0)).parent()  # defined in subspace.py
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.coordinate_vector(subspace.gen(0))           # defined in subspace.py
(1, 0)

degree()

Return the degree of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=4, k=24, ep=-1)
sage: MF.degree()
3
sage: MF.subspace([MF.gen(0), MF.gen(2)]).degree() # defined in subspace.py
3

dimension()

Return the dimension of self.

Note

This method should be overloaded by subclasses.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: QuasiMeromorphicModularForms(k=2, ep=-1).dimension()
+Infinity

element_from_ambient_coordinates(vec)

If self has an associated free module, then return the element of self corresponding to the given vec. Otherwise raise an exception.

INPUT:

• vec – An element of self.module() or self.ambient_module().

OUTPUT:

An element of self corresponding to vec.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(k=24)
sage: MF.dimension()
3
sage: el = MF.element_from_ambient_coordinates([1,1,1])
sage: el == MF.element_from_coordinates([1,1,1])
True
sage: el.parent() == MF
True

sage: subspace = MF.subspace([MF.gen(0), MF.gen(1)])
sage: el = subspace.element_from_ambient_coordinates([1,1,0])
sage: el
1 + q + 52611660*q^3 + 39019412128*q^4 + O(q^5)
sage: el.parent() == subspace
True

element_from_coordinates(vec)

If self has an associated free module, then return the element of self corresponding to the given coordinate vector vec. Otherwise raise an exception.

INPUT:

• vec – A coordinate vector with respect to self.gens().

OUTPUT:

An element of self corresponding to the coordinate vector vec.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(k=24)
sage: MF.dimension()
3
sage: el = MF.element_from_coordinates([1,1,1])
sage: el
1 + q + q^2 + 52611612*q^3 + 39019413208*q^4 + O(q^5)
sage: el == MF.gen(0) + MF.gen(1) + MF.gen(2)
True
sage: el.parent() == MF
True

sage: subspace = MF.subspace([MF.gen(0), MF.gen(1)])
sage: el = subspace.element_from_coordinates([1,1])
sage: el
1 + q + 52611660*q^3 + 39019412128*q^4 + O(q^5)
sage: el == subspace.gen(0) + subspace.gen(1)
True
sage: el.parent() == subspace
True

ep()

Return the multiplier of (elements of) self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: QuasiModularForms(n=16, k=16/7, ep=-1).ep()
-1

faber_pol(m, order_1=0, fix_d=False, d_num_prec=None)

If n=infinity a non-trivial order of -1 can be specified through the parameter order_1 (default: 0). Otherwise it is ignored. Return the \(m\)’th Faber polynomial of self with a different normalization based on j_inv instead of J_inv.

Namely a polynomial p(q) such that p(j_inv)*F_simple() has a Fourier expansion of the form q^m + O(q^(order_inf + 1)). where order_inf = self._l1 - order_1 and p(q) is a monic polynomial of degree order_inf - m.

If n=infinity a non-trivial order of -1 can be specified through the parameter order_1 (default: 0). Otherwise it is ignored.

The relation to Faber_pol is: faber_pol(q) = Faber_pol(d*q).

INPUT:

• m – An integer m <= self._l1 - order_1.

• order_1 – The order at -1 of F_simple (default: 0).

  This parameter is ignored if n != infinity.

• 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 corresponding Faber polynomial p(q).

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms
sage: MF = WeakModularForms(n=5, k=62/3, ep=-1)
sage: MF.weight_parameters()
(2, 3)

sage: MF.faber_pol(2)
1
sage: MF.faber_pol(1)
q - 19/(100*d)
sage: MF.faber_pol(0)
q^2 - 117/(200*d)*q + 9113/(320000*d^2)
sage: MF.faber_pol(-2)
q^4 - 11/(8*d)*q^3 + 41013/(80000*d^2)*q^2 - 2251291/(48000000*d^3)*q + 1974089431/(4915200000000*d^4)
sage: (MF.faber_pol(2)(MF.j_inv())*MF.F_simple()).q_expansion(prec=MF._l1+2)
q^2 - 41/(200*d)*q^3 + O(q^4)
sage: (MF.faber_pol(1)(MF.j_inv())*MF.F_simple()).q_expansion(prec=MF._l1+1)
q + O(q^3)
sage: (MF.faber_pol(0)(MF.j_inv())*MF.F_simple()).q_expansion(prec=MF._l1+1)
1 + O(q^3)
sage: (MF.faber_pol(-2)(MF.j_inv())*MF.F_simple()).q_expansion(prec=MF._l1+1)
q^-2 + O(q^3)

sage: MF = WeakModularForms(n=4, k=-2, ep=1)
sage: MF.weight_parameters()
(-1, 3)

sage: MF.faber_pol(-1)
1
sage: MF.faber_pol(-2, fix_d=True)
q - 184
sage: MF.faber_pol(-3, fix_d=True)
q^2 - 288*q + 14364
sage: (MF.faber_pol(-1, fix_d=True)(MF.j_inv())*MF.F_simple()).q_expansion(prec=MF._l1+2, fix_d=True)
q^-1 + 80 + O(q)
sage: (MF.faber_pol(-2, fix_d=True)(MF.j_inv())*MF.F_simple()).q_expansion(prec=MF._l1+2, fix_d=True)
q^-2 + 400 + O(q)
sage: (MF.faber_pol(-3)(MF.j_inv())*MF.F_simple()).q_expansion(prec=MF._l1+2, fix_d=True)
q^-3 + 2240 + O(q)

sage: MF = WeakModularForms(n=infinity, k=14, ep=-1)
sage: MF.faber_pol(3)
1
sage: MF.faber_pol(2)
q + 3/(8*d)
sage: MF.faber_pol(1)
q^2 + 75/(1024*d^2)
sage: MF.faber_pol(0)
q^3 - 3/(8*d)*q^2 + 3/(512*d^2)*q + 41/(4096*d^3)
sage: MF.faber_pol(-1)
q^4 - 3/(4*d)*q^3 + 81/(1024*d^2)*q^2 + 9075/(8388608*d^4)
sage: (MF.faber_pol(-1)(MF.j_inv())*MF.F_simple()).q_expansion(prec=MF._l1 + 1)
q^-1 + O(q^4)

sage: MF.faber_pol(3, order_1=-1)
q + 3/(4*d)
sage: MF.faber_pol(1, order_1=2)
1
sage: MF.faber_pol(0, order_1=2)
q - 3/(8*d)
sage: MF.faber_pol(-1, order_1=2)
q^2 - 3/(4*d)*q + 81/(1024*d^2)
sage: (MF.faber_pol(-1, order_1=2)(MF.j_inv())*MF.F_simple(order_1=2)).q_expansion(prec=MF._l1 + 1)
q^-1 - 9075/(8388608*d^4)*q^3 + O(q^4)

gen(k=0)

Return the k’th basis element of self if possible (default: k=0).

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: ModularForms(k=12).gen(1).parent()
ModularForms(n=3, k=12, ep=1) over Integer Ring
sage: ModularForms(k=12).gen(1)
q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5)

gens()

This method should be overloaded by subclasses.

Return a basis of self.

Note that the coordinate vector of elements of self are with respect to this basis.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: ModularForms(k=12).gens() # defined in space.py
[1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + O(q^5),
 q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5)]

homogeneous_part(k, ep)

Since self already is a homogeneous component return self unless the degree differs in which case a ValueError is raised.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: MF = QuasiMeromorphicModularForms(n=6, k=4)
sage: MF == MF.homogeneous_part(4,1)
True
sage: MF.homogeneous_part(5,1)
Traceback (most recent call last):
...
ValueError: QuasiMeromorphicModularForms(n=6, k=4, ep=1) over Integer Ring already is homogeneous with degree (4, 1) != (5, 1)!

is_ambient()

Return whether self is an ambient space.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(k=12)
sage: MF.is_ambient()
True
sage: MF.subspace([MF.gen(0)]).is_ambient()
False

module()

Return the module associated to self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(k=12)
sage: MF.module()
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace = MF.subspace([MF.gen(0)])
sage: subspace.module()
Vector space of degree 2 and dimension 1 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
Basis matrix:
[1 0]

one()

Return the one element from the corresponding space of constant forms.

Note

The one element does not lie in self in general.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import CuspForms
sage: MF = CuspForms(k=12)
sage: MF.Delta()^0 == MF.one()
True
sage: (MF.Delta()^0).parent()
ModularForms(n=3, k=0, ep=1) over Integer Ring

q_basis(m=None, min_exp=0, order_1=0)

Try to return a (basis) element of self with a Laurent series of the form q^m + O(q^N), where N=self.required_laurent_prec(min_exp).

If m==None the whole basis (with varying m’s) is returned if it exists.

INPUT:

• m – An integer, indicating the desired initial Laurent exponent of the element.

  If m==None (default) then the whole basis is returned.

• min_exp – An integer, indicating the minimal Laurent exponent (for each quasi part)

  of the subspace of self which should be considered (default: 0).

• order_1 – A lower bound for the order at -1 of all quasi parts of the subspace

  (default: 0). If n!=infinity this parameter is ignored.

OUTPUT:

The corresponding basis (if m==None) resp. the corresponding basis vector (if m!=None). If the basis resp. element doesn’t exist an exception is raised.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiWeakModularForms, ModularForms, QuasiModularForms
sage: QF = QuasiWeakModularForms(n=8, k=10/3, ep=-1)
sage: QF.default_prec(QF.required_laurent_prec(min_exp=-1))
sage: q_basis = QF.q_basis(min_exp=-1)
sage: q_basis
[q^-1 + O(q^5), 1 + O(q^5), q + O(q^5), q^2 + O(q^5), q^3 + O(q^5), q^4 + O(q^5)]
sage: QF.q_basis(m=-1, min_exp=-1)
q^-1 + O(q^5)

sage: MF = ModularForms(k=36)
sage: MF.q_basis() == MF.gens()
True

sage: QF = QuasiModularForms(k=6)
sage: QF.required_laurent_prec()
3
sage: QF.q_basis()
[1 - 20160*q^3 - 158760*q^4 + O(q^5), q - 60*q^3 - 248*q^4 + O(q^5), q^2 + 8*q^3 + 30*q^4 + O(q^5)]

sage: QF = QuasiWeakModularForms(n=infinity, k=-2, ep=-1)
sage: QF.q_basis(order_1=-1)
[1 - 168*q^2 + 2304*q^3 - 19320*q^4 + O(q^5),
 q - 18*q^2 + 180*q^3 - 1316*q^4 + O(q^5)]

quasi_part_dimension(r=None, min_exp=0, max_exp=+Infinity, order_1=0)

Return the dimension of the subspace of self generated by self.quasi_part_gens(r, min_exp, max_exp, order_1).

See quasi_part_gens() for more details.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms, QuasiCuspForms, QuasiWeakModularForms
sage: MF = QuasiModularForms(n=5, k=6, ep=-1)
sage: [v.as_ring_element() for v in MF.gens()]
[f_rho^2*f_i, f_rho^3*E2, E2^3]
sage: MF.dimension()
3
sage: MF.quasi_part_dimension(r=0)
1
sage: MF.quasi_part_dimension(r=1)
1
sage: MF.quasi_part_dimension(r=2)
0
sage: MF.quasi_part_dimension(r=3)
1

sage: MF = QuasiCuspForms(n=5, k=18, ep=-1)
sage: MF.dimension()
8
sage: MF.quasi_part_dimension(r=0)
2
sage: MF.quasi_part_dimension(r=1)
2
sage: MF.quasi_part_dimension(r=2)
1
sage: MF.quasi_part_dimension(r=3)
1
sage: MF.quasi_part_dimension(r=4)
1
sage: MF.quasi_part_dimension(r=5)
1
sage: MF.quasi_part_dimension(min_exp=2, max_exp=2)
2

sage: MF = QuasiCuspForms(n=infinity, k=18, ep=-1)
sage: MF.quasi_part_dimension(r=1, min_exp=-2)
3
sage: MF.quasi_part_dimension()
12
sage: MF.quasi_part_dimension(order_1=3)
2

sage: MF = QuasiWeakModularForms(n=infinity, k=4, ep=1)
sage: MF.quasi_part_dimension(min_exp=2, order_1=-2)
4
sage: [v.order_at(-1) for v in MF.quasi_part_gens(r=0, min_exp=2, order_1=-2)]
[-2, -2]

quasi_part_gens(r=None, min_exp=0, max_exp=+Infinity, order_1=0)

Return a basis in self of the subspace of (quasi) weakly holomorphic forms which satisfy the specified properties on the quasi parts and the initial Fourier coefficient.

INPUT:

• r – An integer or None (default), indicating

  the desired power of E2 If r=None then all possible powers (r) are choosen.

• min_exp – An integer giving a lower bound for the

  first non-trivial Fourier coefficient of the generators (default: 0).

• max_exp – An integer or infinity (default) giving

  an upper bound for the first non-trivial Fourier coefficient of the generators. If max_exp==infinity then no upper bound is assumed.

• order_1 – A lower bound for the order at -1 of all

  quasi parts of the basis elements (default: 0). If n!=infinity this parameter is ignored.

OUTPUT:

A basis in self of the subspace of forms which are modular after dividing by E2^r and which have a Fourier expansion of the form q^m + O(q^(m+1)) with min_exp <= m <= max_exp for each quasi part (and at least the specified order at -1 in case n=infinity). Note that linear combinations of forms/quasi parts maybe have a higher order at infinity than max_exp.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiWeakModularForms
sage: QF = QuasiWeakModularForms(n=8, k=10/3, ep=-1)
sage: QF.default_prec(1)
sage: QF.quasi_part_gens(min_exp=-1)
[q^-1 + O(q), 1 + O(q), q^-1 - 9/(128*d) + O(q), 1 + O(q), q^-1 - 19/(64*d) + O(q), q^-1 + 1/(64*d) + O(q)]

sage: QF.quasi_part_gens(min_exp=-1, max_exp=-1)
[q^-1 + O(q), q^-1 - 9/(128*d) + O(q), q^-1 - 19/(64*d) + O(q), q^-1 + 1/(64*d) + O(q)]
sage: QF.quasi_part_gens(min_exp=-2, r=1)
[q^-2 - 9/(128*d)*q^-1 - 261/(131072*d^2) + O(q), q^-1 - 9/(128*d) + O(q), 1 + O(q)]

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(k=36)
sage: MF.quasi_part_gens(min_exp=2)
[q^2 + 194184*q^4 + O(q^5), q^3 - 72*q^4 + O(q^5)]

sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: MF = QuasiModularForms(n=5, k=6, ep=-1)
sage: MF.default_prec(2)
sage: MF.dimension()
3
sage: MF.quasi_part_gens(r=0)
[1 - 37/(200*d)*q + O(q^2)]
sage: MF.quasi_part_gens(r=0)[0] == MF.E6()
True
sage: MF.quasi_part_gens(r=1)
[1 + 33/(200*d)*q + O(q^2)]
sage: MF.quasi_part_gens(r=1)[0] == MF.E2()*MF.E4()
True
sage: MF.quasi_part_gens(r=2)
[]
sage: MF.quasi_part_gens(r=3)
[1 - 27/(200*d)*q + O(q^2)]
sage: MF.quasi_part_gens(r=3)[0] == MF.E2()^3
True

sage: from sage.modular.modform_hecketriangle.space import QuasiCuspForms, CuspForms
sage: MF = QuasiCuspForms(n=5, k=18, ep=-1)
sage: MF.default_prec(4)
sage: MF.dimension()
8
sage: MF.quasi_part_gens(r=0)
[q - 34743/(640000*d^2)*q^3 + O(q^4), q^2 - 69/(200*d)*q^3 + O(q^4)]
sage: MF.quasi_part_gens(r=1)
[q - 9/(200*d)*q^2 + 37633/(640000*d^2)*q^3 + O(q^4),
 q^2 + 1/(200*d)*q^3 + O(q^4)]
sage: MF.quasi_part_gens(r=2)
[q - 1/(4*d)*q^2 - 24903/(640000*d^2)*q^3 + O(q^4)]
sage: MF.quasi_part_gens(r=3)
[q + 1/(10*d)*q^2 - 7263/(640000*d^2)*q^3 + O(q^4)]
sage: MF.quasi_part_gens(r=4)
[q - 11/(20*d)*q^2 + 53577/(640000*d^2)*q^3 + O(q^4)]
sage: MF.quasi_part_gens(r=5)
[q - 1/(5*d)*q^2 + 4017/(640000*d^2)*q^3 + O(q^4)]

sage: MF.quasi_part_gens(r=1)[0] == MF.E2() * CuspForms(n=5, k=16, ep=1).gen(0)
True
sage: MF.quasi_part_gens(r=1)[1] == MF.E2() * CuspForms(n=5, k=16, ep=1).gen(1)
True
sage: MF.quasi_part_gens(r=3)[0] == MF.E2()^3 * MF.Delta()
True

sage: MF = QuasiCuspForms(n=infinity, k=18, ep=-1)
sage: MF.quasi_part_gens(r=1, min_exp=-2) == MF.quasi_part_gens(r=1, min_exp=1)
True
sage: MF.quasi_part_gens(r=1)
[q - 8*q^2 - 8*q^3 + 5952*q^4 + O(q^5),
 q^2 - 8*q^3 + 208*q^4 + O(q^5),
 q^3 - 16*q^4 + O(q^5)]

sage: MF = QuasiWeakModularForms(n=infinity, k=4, ep=1)
sage: MF.quasi_part_gens(r=2, min_exp=2, order_1=-2)[0] == MF.E2()^2 * MF.E4()^(-2) * MF.f_inf()^2
True
sage: [v.order_at(-1) for v in MF.quasi_part_gens(r=0, min_exp=2, order_1=-2)]
[-2, -2]

rank()

Return the rank of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=4, k=24, ep=-1)
sage: MF.rank()
3
sage: MF.subspace([MF.gen(0), MF.gen(2)]).rank()
2

rationalize_series(laurent_series, coeff_bound=1e-10, denom_factor=1)

Try to return a Laurent series with coefficients in self.coeff_ring() that matches the given Laurent series.

We give our best but there is absolutely no guarantee that it will work!

INPUT:

• laurent_series – A Laurent series. If the Laurent coefficients already

  coerce into self.coeff_ring() with a formal parameter then the Laurent series is returned as is.

  Otherwise it is assumed that the series is normalized in the sense that the first non-trivial coefficient is a power of d (e.g. 1).

• coeff_bound – Either None resp. 0 or a positive real number

  (default: 1e-10). If specified coeff_bound gives a lower bound for the size of the initial Laurent coefficients. If a coefficient is smaller it is assumed to be zero.

  For calculations with very small coefficients (less than 1e-10) coeff_bound should be set to something even smaller or just 0.

  Non-exact calculations often produce non-zero coefficients which are supposed to be zero. In those cases this parameter helps a lot.

• denom_factor – An integer (default: 1) whose factor might occur in

  the denominator of the given Laurent coefficients (in addition to naturally occuring factors).

OUTPUT:

A Laurent series over self.coeff_ring() corresponding to the given Laurent series.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import WeakModularForms, ModularForms, QuasiCuspForms
sage: WF = WeakModularForms(n=14)
sage: qexp = WF.J_inv().q_expansion_fixed_d(d_num_prec=1000)
sage: qexp.parent()
Laurent Series Ring in q over Real Field with 1000 bits of precision
sage: qexp_int = WF.rationalize_series(qexp)
sage: qexp_int.add_bigoh(3)
d*q^-1 + 37/98 + 2587/(38416*d)*q + 899/(117649*d^2)*q^2 + O(q^3)
sage: qexp_int == WF.J_inv().q_expansion()
True
sage: WF.rationalize_series(qexp_int) == qexp_int
True
sage: WF(qexp_int) == WF.J_inv()
True

sage: WF.rationalize_series(qexp.parent()(1))
1
sage: WF.rationalize_series(qexp_int.parent()(1)).parent()
Laurent Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring

sage: MF = ModularForms(n=infinity, k=4)
sage: qexp = MF.E4().q_expansion_fixed_d()
sage: qexp.parent()
Power Series Ring in q over Rational Field
sage: qexp_int = MF.rationalize_series(qexp)
sage: qexp_int.parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: qexp_int == MF.E4().q_expansion()
True
sage: MF.rationalize_series(qexp_int) == qexp_int
True
sage: MF(qexp_int) == MF.E4()
True

sage: QF = QuasiCuspForms(n=8, k=22/3, ep=-1)
sage: el = QF(QF.f_inf()*QF.E2())
sage: qexp = el.q_expansion_fixed_d(d_num_prec=1000)
sage: qexp.parent()
Power Series Ring in q over Real Field with 1000 bits of precision
sage: qexp_int = QF.rationalize_series(qexp)
sage: qexp_int.parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: qexp_int == el.q_expansion()
True
sage: QF.rationalize_series(qexp_int) == qexp_int
True
sage: QF(qexp_int) == el
True

required_laurent_prec(min_exp=0, order_1=0)

Return an upper bound for the required precision for Laurent series to uniquely determine a corresponding (quasi) form in self with the given lower bound min_exp for the order at infinity (for each quasi part).

Note

For n=infinity only the holomorphic case (min_exp >= 0) is supported (in particular a non-negative order at -1 is assumed).

INPUT:

• min_exp – An integer (default: 0), namely the lower bound for the

  order at infinity resp. the exponent of the Laurent series.

• order_1 – A lower bound for the order at -1 for all quasi parts

  (default: 0). If n!=infinity this parameter is ignored.

OUTPUT:

An integer, namely an upper bound for the number of required Laurent coefficients. The bound should be precise or at least pretty sharp.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiWeakModularForms, ModularForms, QuasiModularForms
sage: QF = QuasiWeakModularForms(n=8, k=10/3, ep=-1)
sage: QF.required_laurent_prec(min_exp=-1)
5

sage: MF = ModularForms(k=36)
sage: MF.required_laurent_prec(min_exp=2)
4

sage: QuasiModularForms(k=2).required_laurent_prec()
1

sage: QuasiWeakModularForms(n=infinity, k=2, ep=-1).required_laurent_prec(order_1=-1)
6

subspace(basis)

Return the subspace of self generated by basis.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(k=24)
sage: MF.dimension()
3
sage: subspace = MF.subspace([MF.gen(0), MF.gen(1)])
sage: subspace
Subspace of dimension 2 of ModularForms(n=3, k=24, ep=1) over Integer Ring

weight()

Return the weight of (elements of) self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: QuasiModularForms(n=16, k=16/7, ep=-1).weight()
16/7

weight_parameters()

Check whether self has a valid weight and multiplier.

If not then an exception is raised. Otherwise the two weight parameters corresponding to the weight and multiplier of self are returned.

The weight parameters are e.g. used to calculate dimensions or precisions of Fourier expansion.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import MeromorphicModularForms
sage: MF = MeromorphicModularForms(n=18, k=-7, ep=-1)
sage: MF.weight_parameters()
(-3, 17)
sage: (MF._l1, MF._l2) == MF.weight_parameters()
True
sage: (k, ep) = (MF.weight(), MF.ep())
sage: n = MF.hecke_n()
sage: k == 4*(n*MF._l1 + MF._l2)/(n-2) + (1-ep)*n/(n-2)
True

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=5, k=12, ep=1)
sage: MF.weight_parameters()
(1, 4)
sage: (MF._l1, MF._l2) == MF.weight_parameters()
True
sage: (k, ep) = (MF.weight(), MF.ep())
sage: n = MF.hecke_n()
sage: k == 4*(n*MF._l1 + MF._l2)/(n-2) + (1-ep)*n/(n-2)
True
sage: MF.dimension() == MF._l1 + 1
True

sage: MF = ModularForms(n=infinity, k=8, ep=1)
sage: MF.weight_parameters()
(2, 0)
sage: MF.dimension() == MF._l1 + 1
True