Elements of graded rings of modular forms for Hecke triangle groups

AUTHORS:

• Jonas Jermann (2013): initial version

class sage.modular.modform_hecketriangle.graded_ring_element.FormsRingElement(parent, rat)

Bases: sage.structure.element.CommutativeAlgebraElement, sage.structure.unique_representation.UniqueRepresentation

Element of a FormsRing.

AT = Analytic Type

AnalyticType

alias of sage.modular.modform_hecketriangle.analytic_type.AnalyticType

analytic_type()

Return the analytic type of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiMeromorphicModularFormsRing(n=5)(x/z+d).analytic_type()
quasi meromorphic modular
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).analytic_type()
quasi weakly holomorphic modular
sage: QuasiMeromorphicModularFormsRing(n=5)(x^2+y-d).analytic_type()
modular
sage: QuasiMeromorphicModularForms(n=18).J_inv().analytic_type()
weakly holomorphic modular
sage: QuasiMeromorphicModularForms(n=18).f_inf().analytic_type()
cuspidal
sage: QuasiMeromorphicModularForms(n=infinity).f_inf().analytic_type()
modular

as_ring_element()

Coerce self into the graded ring of its parent.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import CuspForms
sage: Delta = CuspForms(k=12).Delta()
sage: Delta.parent()
CuspForms(n=3, k=12, ep=1) over Integer Ring
sage: Delta.as_ring_element()
f_rho^3*d - f_i^2*d
sage: Delta.as_ring_element().parent()
CuspFormsRing(n=3) over Integer Ring

sage: CuspForms(n=infinity, k=12).Delta().as_ring_element()
-E4^2*f_i^2*d + E4^3*d

base_ring()

Return base ring of self.parent().

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: ModularForms(n=12, k=4, base_ring=CC).E4().base_ring()
Complex Field with 53 bits of precision

coeff_ring()

Return coefficient ring of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
sage: ModularFormsRing().E6().coeff_ring()
Fraction Field of Univariate Polynomial Ring in d over Integer Ring

degree()

Return the degree of self in the graded ring. If self is not homogeneous, then (None, None) is returned.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiModularFormsRing()(x+y).degree() == (None, None)
True
sage: ModularForms(n=18).f_i().degree()
(9/4, -1)
sage: ModularForms(n=infinity).f_rho().degree()
(0, 1)

denominator()

Return the denominator of self. I.e. the (properly reduced) new form corresponding to the numerator of self.rat().

Note that the parent of self might (probably will) change.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiMeromorphicModularFormsRing(n=5).Delta().full_reduce().denominator()
1 + O(q^5)
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).denominator()
f_rho^5 - f_i^2
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).denominator().parent()
QuasiModularFormsRing(n=5) over Integer Ring
sage: QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).denominator()
1 - 13/(40*d)*q - 351/(64000*d^2)*q^2 - 13819/(76800000*d^3)*q^3 - 1163669/(491520000000*d^4)*q^4 + O(q^5)
sage: QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).denominator().parent()
QuasiModularForms(n=5, k=10/3, ep=-1) over Integer Ring
sage: (QuasiMeromorphicModularForms(n=infinity, k=-6, ep=-1)(y/(x*(x-y^2)))).denominator()
-64*q - 512*q^2 - 768*q^3 + 4096*q^4 + O(q^5)
sage: (QuasiMeromorphicModularForms(n=infinity, k=-6, ep=-1)(y/(x*(x-y^2)))).denominator().parent()
QuasiModularForms(n=+Infinity, k=8, ep=1) over Integer Ring

derivative()

Return the derivative d/dq = lambda/(2*pi*i) d/dtau of self.

Note that the parent might (probably will) change. In particular its analytic type will be extended to contain “quasi”.

If parent.has_reduce_hom() == True then the result is reduced to be an element of the corresponding forms space if possible.

In particular this is the case if self is a (homogeneous) element of a forms space.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(n=7, red_hom=True)
sage: n = MR.hecke_n()
sage: E2 = MR.E2().full_reduce()
sage: E6 = MR.E6().full_reduce()
sage: f_rho = MR.f_rho().full_reduce()
sage: f_i = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()

sage: derivative(f_rho) == 1/n * (f_rho*E2 - f_i)
True
sage: derivative(f_i)   == 1/2 * (f_i*E2 - f_rho**(n-1))
True
sage: derivative(f_inf) == f_inf * E2
True
sage: derivative(f_inf).parent()
QuasiCuspForms(n=7, k=38/5, ep=-1) over Integer Ring
sage: derivative(E2)    == (n-2)/(4*n) * (E2**2 - f_rho**(n-2))
True
sage: derivative(E2).parent()
QuasiModularForms(n=7, k=4, ep=1) over Integer Ring

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: E2 = MR.E2().full_reduce()
sage: E4 = MR.E4().full_reduce()
sage: E6 = MR.E6().full_reduce()
sage: f_i = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()

sage: derivative(E4)    == E4 * (E2 - f_i)
True
sage: derivative(f_i)   == 1/2 * (f_i*E2 - E4)
True
sage: derivative(f_inf) == f_inf * E2
True
sage: derivative(f_inf).parent()
QuasiModularForms(n=+Infinity, k=6, ep=-1) over Integer Ring
sage: derivative(E2)    == 1/4 * (E2**2 - E4)
True
sage: derivative(E2).parent()
QuasiModularForms(n=+Infinity, k=4, ep=1) over Integer Ring

diff_op(op, new_parent=None)

Return the differential operator op applied to self. If parent.has_reduce_hom() == True then the result is reduced to be an element of the corresponding forms space if possible.

INPUT:

• op – An element of self.parent().diff_alg().

  I.e. an element of the algebra over QQ of differential operators generated by X, Y, Z, dX, dY, DZ, where e.g. X corresponds to the multiplication by x (resp. f_rho) and dX corresponds to d/dx.

  To expect a homogeneous result after applying the operator to a homogeneous element it should should be homogeneous operator (with respect to the usual, special grading).

• new_parent – Try to convert the result to the specified

  new_parent. If new_parent == None (default) then the parent is extended to a “quasi meromorphic” ring.

OUTPUT:

The new element.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(n=8, red_hom=True)
sage: (X,Y,Z,dX,dY,dZ) = MR.diff_alg().gens()
sage: n=MR.hecke_n()
sage: mul_op = 4/(n-2)*X*dX + 2*n/(n-2)*Y*dY + 2*Z*dZ
sage: der_op = MR._derivative_op()
sage: ser_op = MR._serre_derivative_op()
sage: der_op == ser_op + (n-2)/(4*n)*Z*mul_op
True

sage: Delta = MR.Delta().full_reduce()
sage: E2 = MR.E2().full_reduce()
sage: Delta.diff_op(mul_op) == 12*Delta
True
sage: Delta.diff_op(mul_op).parent()
QuasiMeromorphicModularForms(n=8, k=12, ep=1) over Integer Ring
sage: Delta.diff_op(mul_op, Delta.parent()).parent()
CuspForms(n=8, k=12, ep=1) over Integer Ring
sage: E2.diff_op(mul_op, E2.parent()) == 2*E2
True
sage: Delta.diff_op(Z*mul_op, Delta.parent().extend_type("quasi", ring=True)) == 12*E2*Delta
True

sage: ran_op = X + Y*X*dY*dX + dZ + dX^2
sage: Delta.diff_op(ran_op)
f_rho^19*d + 306*f_rho^16*d - f_rho^11*f_i^2*d - 20*f_rho^10*f_i^2*d - 90*f_rho^8*f_i^2*d
sage: E2.diff_op(ran_op)
f_rho*E2 + 1

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: (X,Y,Z,dX,dY,dZ) = MR.diff_alg().gens()
sage: mul_op = 4*X*dX + 2*Y*dY + 2*Z*dZ
sage: der_op = MR._derivative_op()
sage: ser_op = MR._serre_derivative_op()
sage: der_op == ser_op + Z/4*mul_op
True

sage: Delta = MR.Delta().full_reduce()
sage: E2 = MR.E2().full_reduce()
sage: Delta.diff_op(mul_op) == 12*Delta
True
sage: Delta.diff_op(mul_op).parent()
QuasiMeromorphicModularForms(n=+Infinity, k=12, ep=1) over Integer Ring
sage: Delta.diff_op(mul_op, Delta.parent()).parent()
CuspForms(n=+Infinity, k=12, ep=1) over Integer Ring
sage: E2.diff_op(mul_op, E2.parent()) == 2*E2
True
sage: Delta.diff_op(Z*mul_op, Delta.parent().extend_type("quasi", ring=True)) == 12*E2*Delta
True

sage: ran_op = X + Y*X*dY*dX + dZ + dX^2
sage: Delta.diff_op(ran_op)
-E4^3*f_i^2*d + E4^4*d - 4*E4^2*f_i^2*d - 2*f_i^2*d + 6*E4*d
sage: E2.diff_op(ran_op)
E4*E2 + 1

ep()

Return the multiplier of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiModularFormsRing()(x+y).ep() is None
True
sage: ModularForms(n=18).f_i().ep()
-1
sage: ModularForms(n=infinity).E2().ep()
-1

evaluate(tau, prec=None, num_prec=None, check=False)

Try to return self evaluated at a point tau in the upper half plane, where self is interpreted as a function in tau, where q=exp(2*pi*i*tau).

Note that this interpretation might not make sense (and fail) for certain (many) choices of (base_ring, tau.parent()).

It is possible to evaluate at points of HyperbolicPlane(). In this case the coordinates of the upper half plane model are used.

To obtain a precise and fast result the parameters prec and num_prec both have to be considered/balanced. A high prec value is usually quite costly.

INPUT:

• tau – infinity or an element of the upper

  half plane. E.g. with parent AA or CC.

• prec – An integer, namely the precision used for the

  Fourier expansion. If prec == None (default) then the default precision of self.parent() is used.

• num_prec – An integer, namely the minimal numerical precision

  used for tau and d. If num_prec == None (default) then the default numerical precision of self.parent() is used.

• check – If True then the order of tau is checked.

  Otherwise the order is only considered for tau = infinity, i, rho, -1/rho. Default: False.

OUTPUT:

The (numerical) evaluated function value.

ALGORITHM:

1. If the order of self at tau is known and nonzero: Return 0 resp. infinity.

2. Else if tau==infinity and the order is zero: Return the constant Fourier coefficient of self.

3. Else if self is homogeneous and modular:

   1. Because of the (modular) transformation property of self the evaluation at tau is given by the evaluation at w multiplied by aut_factor(A,w).
   2. The evaluation at w is calculated by evaluating the truncated Fourier expansion of self at q(w).

   Note that this is much faster and more precise than a direct evaluation at tau.

4. Else if self is exactly E2:

   1. The same procedure as before is applied (with the aut_factor from the corresponding modular space).
   2. Except that at the end a correction term for the quasimodular form E2 of the form 4*lambda/(2*pi*i)*n/(n-2) * c*(c*w + d) (resp. 4/(pi*i) * c*(c*w + d) for n=infinity) has to be added, where lambda = 2*cos(pi/n) (resp lambda = 2 for n=infinity) and c,d are the lower entries of the matrix A.

5. Else:

   1. Evaluate f_rho, f_i, E2 at tau using the above procedures. If n=infinity use E4 instead of f_rho.
   2. Substitute x=f_rho(tau), y=f_i(tau), z=E2(tau) and the numerical value of d for d in self.rat(). If n=infinity then substitute x=E4(tau) instead.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(n=5, red_hom=True)
sage: f_rho = MR.f_rho().full_reduce()
sage: f_i   = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()
sage: E2    = MR.E2().full_reduce()
sage: E4    = MR.E4().full_reduce()
sage: rho   = MR.group().rho()

sage: f_rho(rho)
0
sage: f_rho(rho + 1e-100)    # since rho == rho + 1e-100
0
sage: f_rho(rho + 1e-6)
2.525...e-10 - 3.884...e-6*I
sage: f_i(i)
0
sage: f_i(i + 1e-1000)  # rel tol 5e-2
-6.08402217494586e-14 - 4.10147008296517e-1000*I
sage: f_inf(infinity)
0

sage: i = I = QuadraticField(-1, 'I').gen()
sage: z = -1/(-1/(2*i+30)-1)
sage: z
2/965*I + 934/965
sage: E4(z)
32288.05588811... - 118329.8566016...*I
sage: E4(z, prec=30, num_prec=100)    # long time
32288.0558872351130041311053... - 118329.856600349999751420381...*I
sage: E2(z)
409.3144737105... + 100.6926857489...*I
sage: E2(z, prec=30, num_prec=100)    # long time
409.314473710489761254584951... + 100.692685748952440684513866...*I
sage: (E2^2-E4)(z)
125111.2655383... + 200759.8039479...*I
sage: (E2^2-E4)(z, prec=30, num_prec=100)    # long time
125111.265538336196262200469... + 200759.803948009905410385699...*I

sage: (E2^2-E4)(infinity)
0
sage: (1/(E2^2-E4))(infinity)
+Infinity
sage: ((E2^2-E4)/f_inf)(infinity)
-3/(10*d)

sage: G = HeckeTriangleGroup(n=8)
sage: MR = QuasiMeromorphicModularFormsRing(group=G, red_hom=True)
sage: f_rho = MR.f_rho().full_reduce()
sage: f_i   = MR.f_i().full_reduce()
sage: E2    = MR.E2().full_reduce()

sage: z = AlgebraicField()(1/10+13/10*I)
sage: A = G.V(4)
sage: S = G.S()
sage: T = G.T()
sage: A == (T*S)**3*T
True
sage: az = A.acton(z)
sage: az == (A[0,0]*z + A[0,1]) / (A[1,0]*z + A[1,1])
True

sage: f_rho(z)
1.03740476727... + 0.0131941034523...*I
sage: f_rho(az)
-2.29216470688... - 1.46235057536...*I
sage: k = f_rho.weight()
sage: aut_fact = f_rho.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f_rho.parent().aut_factor(A, z)) < 1e-12
True
sage: aut_fact * f_rho(z)
-2.29216470688... - 1.46235057536...*I

sage: f_rho.parent().default_num_prec(1000)
sage: f_rho.parent().default_prec(300)
sage: (f_rho.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
1.0374047672719462149821251... + 0.013194103452368974597290332...*I
sage: (f_rho.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
-2.2921647068881834598616367... - 1.4623505753697635207183406...*I

sage: f_i(z)
0.667489320423... - 0.118902824870...*I
sage: f_i(az)
14.5845388476... - 28.4604652892...*I
sage: k = f_i.weight()
sage: aut_fact = f_i.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f_i.parent().aut_factor(A, z)) < 1e-12
True
sage: aut_fact * f_i(z)
14.5845388476... - 28.4604652892...*I

sage: f_i.parent().default_num_prec(1000)
sage: f_i.parent().default_prec(300)
sage: (f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
0.66748932042300250077433252... - 0.11890282487028677063054267...*I
sage: (f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
14.584538847698600875918891... - 28.460465289220303834894855...*I

sage: f = f_rho*E2
sage: f(z)
0.966024386418... - 0.0138894699429...*I
sage: f(az)
-15.9978074989... - 29.2775758341...*I
sage: k = f.weight()
sage: aut_fact = f.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f.parent().aut_factor(A, z)) < 1e-12
True
sage: k2 = f_rho.weight()
sage: aut_fact2 = f_rho.ep() * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k2 * (((T*S)*T).acton(z)/AlgebraicField()(i))**k2 * (T.acton(z)/AlgebraicField()(i))**k2
sage: abs(aut_fact2 - f_rho.parent().aut_factor(A, z)) < 1e-12
True
sage: cor_term = (4 * G.n() / (G.n()-2) * A.c() * (A.c()*z+A.d())) / (2*pi*i).n(1000) * G.lam()
sage: aut_fact*f(z) + cor_term*aut_fact2*f_rho(z)
-15.9978074989... - 29.2775758341...*I

sage: f.parent().default_num_prec(1000)
sage: f.parent().default_prec(300)
sage: (f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
0.96602438641867296777809436... - 0.013889469942995530807311503...*I
sage: (f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
-15.997807498958825352887040... - 29.277575834123246063432206...*I

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: f_i   = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()
sage: E2    = MR.E2().full_reduce()
sage: E4    = MR.E4().full_reduce()

sage: f_i(i)
0
sage: f_i(i + 1e-1000)
2.991...e-12 - 3.048...e-1000*I
sage: f_inf(infinity)
0

sage: z = -1/(-1/(2*i+30)-1)
sage: E4(z, prec=15)
804.0722034... + 211.9278206...*I
sage: E4(z, prec=30, num_prec=100)    # long time
803.928382417... + 211.889914044...*I
sage: E2(z)
2.438455612... - 39.48442265...*I
sage: E2(z, prec=30, num_prec=100)    # long time
2.43968197227756036957475... - 39.4842637577742677851431...*I
sage: (E2^2-E4)(z)
-2265.442515... - 380.3197877...*I
sage: (E2^2-E4)(z, prec=30, num_prec=100)    # long time
-2265.44251550679807447320... - 380.319787790548788238792...*I

sage: (E2^2-E4)(infinity)
0
sage: (1/(E2^2-E4))(infinity)
+Infinity
sage: ((E2^2-E4)/f_inf)(infinity)
-1/(2*d)

sage: G = HeckeTriangleGroup(n=Infinity)
sage: z = AlgebraicField()(1/10+13/10*I)
sage: A = G.V(4)
sage: S = G.S()
sage: T = G.T()
sage: A == (T*S)**3*T
True
sage: az = A.acton(z)
sage: az == (A[0,0]*z + A[0,1]) / (A[1,0]*z + A[1,1])
True

sage: f_i(z)
0.6208853409... - 0.1212525492...*I
sage: f_i(az)
6.103314419... + 20.42678597...*I
sage: k = f_i.weight()
sage: aut_fact = f_i.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f_i.parent().aut_factor(A, z)) < 1e-12
True
sage: aut_fact * f_i(z)
6.103314419... + 20.42678597...*I

sage: f_i.parent().default_num_prec(1000)
sage: f_i.parent().default_prec(300)
sage: (f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
0.620885340917559158572271... - 0.121252549240996430425967...*I
sage: (f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
6.10331441975198186745017... + 20.4267859728657976382684...*I

sage: f = f_i*E2
sage: f(z)
0.5349190275... - 0.1322370856...*I
sage: f(az)
-140.4711702... + 469.0793692...*I
sage: k = f.weight()
sage: aut_fact = f.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f.parent().aut_factor(A, z)) < 1e-12
True
sage: k2 = f_i.weight()
sage: aut_fact2 = f_i.ep() * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k2 * (((T*S)*T).acton(z)/AlgebraicField()(i))**k2 * (T.acton(z)/AlgebraicField()(i))**k2
sage: abs(aut_fact2 - f_i.parent().aut_factor(A, z)) < 1e-12
True
sage: cor_term = (4 * A.c() * (A.c()*z+A.d())) / (2*pi*i).n(1000) * G.lam()
sage: aut_fact*f(z) + cor_term*aut_fact2*f_i(z)
-140.4711702... + 469.0793692...*I

sage: f.parent().default_num_prec(1000)
sage: f.parent().default_prec(300)
sage: (f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
0.534919027587592616802582... - 0.132237085641931661668338...*I

sage: (f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
-140.471170232432551196978... + 469.079369280804086032719...*I

It is possible to evaluate at points of HyperbolicPlane():

sage: p = HyperbolicPlane().PD().get_point(-I/2)
sage: bool(p.to_model('UHP').coordinates() == I/3)
True
sage: E4(p) == E4(I/3)
True
sage: p = HyperbolicPlane().PD().get_point(I)
sage: f_inf(p, check=True) == 0
True
sage: (1/(E2^2-E4))(p) == infinity
True

full_reduce()

Convert self into its reduced parent.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: Delta = QuasiMeromorphicModularFormsRing().Delta()
sage: Delta
f_rho^3*d - f_i^2*d
sage: Delta.full_reduce()
q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5)
sage: Delta.full_reduce().parent() == Delta.reduced_parent()
True

sage: QuasiMeromorphicModularFormsRing().Delta().full_reduce().parent()
CuspForms(n=3, k=12, ep=1) over Integer Ring

group()

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

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: ModularForms(n=12, k=4).E4().group()
Hecke triangle group for n = 12

hecke_n()

Return the parameter n of the (Hecke triangle) group of self.parent().

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: ModularForms(n=12, k=6).E6().hecke_n()
12

is_cuspidal()

Return whether self is cuspidal in the sense that self is holomorphic and f_inf divides the numerator.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiModularFormsRing(n=5)(y^3-z^5).is_cuspidal()
False
sage: QuasiModularFormsRing(n=5)(z*x^5-z*y^2).is_cuspidal()
True
sage: QuasiModularForms(n=18).Delta().is_cuspidal()
True
sage: QuasiModularForms(n=18).f_rho().is_cuspidal()
False
sage: QuasiModularForms(n=infinity).f_inf().is_cuspidal()
False
sage: QuasiModularForms(n=infinity).Delta().is_cuspidal()
True

is_holomorphic()

Return whether self is holomorphic in the sense that the denominator of self is constant.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).is_holomorphic()
False
sage: QuasiMeromorphicModularFormsRing(n=5)(x^2+y-d+z).is_holomorphic()
True
sage: QuasiMeromorphicModularForms(n=18).J_inv().is_holomorphic()
False
sage: QuasiMeromorphicModularForms(n=18).f_i().is_holomorphic()
True
sage: QuasiMeromorphicModularForms(n=infinity).f_inf().is_holomorphic()
True

is_homogeneous()

Return whether self is homogeneous.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: QuasiModularFormsRing(n=12).Delta().is_homogeneous()
True
sage: QuasiModularFormsRing(n=12).Delta().parent().is_homogeneous()
False
sage: x,y,z,d=var("x,y,z,d")
sage: QuasiModularFormsRing(n=12)(x^3+y^2+z+d).is_homogeneous()
False

sage: QuasiModularFormsRing(n=infinity)(x*(x-y^2)+y^4).is_homogeneous()
True

is_modular()

Return whether self (resp. its homogeneous components) transform like modular forms.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiModularFormsRing(n=5)(x^2+y-d).is_modular()
True
sage: QuasiModularFormsRing(n=5)(x^2+y-d+z).is_modular()
False
sage: QuasiModularForms(n=18).f_i().is_modular()
True
sage: QuasiModularForms(n=18).E2().is_modular()
False
sage: QuasiModularForms(n=infinity).f_inf().is_modular()
True

is_weakly_holomorphic()

Return whether self is weakly holomorphic in the sense that: self has at most a power of f_inf in its denominator.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiMeromorphicModularFormsRing(n=5)(x/(x^5-y^2)+z).is_weakly_holomorphic()
True
sage: QuasiMeromorphicModularFormsRing(n=5)(x^2+y/x-d).is_weakly_holomorphic()
False
sage: QuasiMeromorphicModularForms(n=18).J_inv().is_weakly_holomorphic()
True
sage: QuasiMeromorphicModularForms(n=infinity, k=-4)(1/x).is_weakly_holomorphic()
True
sage: QuasiMeromorphicModularForms(n=infinity, k=-2)(1/y).is_weakly_holomorphic()
False

is_zero()

Return whether self is the zero function.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiModularFormsRing(n=5)(1).is_zero()
False
sage: QuasiModularFormsRing(n=5)(0).is_zero()
True
sage: QuasiModularForms(n=18).zero().is_zero()
True
sage: QuasiModularForms(n=18).Delta().is_zero()
False
sage: QuasiModularForms(n=infinity).f_rho().is_zero()
False

numerator()

Return the numerator of self.

I.e. the (properly reduced) new form corresponding to the numerator of self.rat().

Note that the parent of self might (probably will) change.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).numerator()
f_rho^5*f_i - f_rho^5*d - E2^5 + f_i^2*d
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).numerator().parent()
QuasiModularFormsRing(n=5) over Integer Ring
sage: QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).numerator()
1 + 7/(100*d)*q + 21/(160000*d^2)*q^2 + 1043/(192000000*d^3)*q^3 + 45479/(1228800000000*d^4)*q^4 + O(q^5)
sage: QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).numerator().parent()
QuasiModularForms(n=5, k=4/3, ep=1) over Integer Ring
sage: (QuasiMeromorphicModularForms(n=infinity, k=-2, ep=-1)(y/x)).numerator()
1 - 24*q + 24*q^2 - 96*q^3 + 24*q^4 + O(q^5)
sage: (QuasiMeromorphicModularForms(n=infinity, k=-2, ep=-1)(y/x)).numerator().parent()
QuasiModularForms(n=+Infinity, k=2, ep=-1) over Integer Ring

order_at(tau=+Infinity)

Return the (overall) order of self at tau if easily possible: Namely if tau is infinity or congruent to i resp. rho.

It is possible to determine the order of points from HyperbolicPlane(). In this case the coordinates of the upper half plane model are used.

If self is homogeneous and modular then the rational function self.rat() is used. Otherwise only tau=infinity is supported by using the Fourier expansion with increasing precision (until the order can be determined).

The function is mainly used to be able to work with the correct precision for Laurent series.

Note

For quasi forms one cannot deduce the analytic type from this order at infinity since the analytic order is defined by the behavior on each quasi part and not by their linear combination.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(red_hom=True)
sage: (MR.Delta()^3).order_at(infinity)
3
sage: MR.E2().order_at(infinity)
0
sage: (MR.J_inv()^2).order_at(infinity)
-2
sage: x,y,z,d = MR.pol_ring().gens()
sage: el = MR((z^3-y)^2/(x^3-y^2)).full_reduce()
sage: el
108*q + 11664*q^2 + 502848*q^3 + 12010464*q^4 + O(q^5)
sage: el.order_at(infinity)
1
sage: el.parent()
QuasiWeakModularForms(n=3, k=0, ep=1) over Integer Ring
sage: el.is_holomorphic()
False
sage: MR((z-y)^2+(x-y)^3).order_at(infinity)
2
sage: MR((x-y)^10).order_at(infinity)
10
sage: MR.zero().order_at(infinity)
+Infinity
sage: (MR(x*y^2)/MR.J_inv()).order_at(i)
2
sage: (MR(x*y^2)/MR.J_inv()).order_at(MR.group().rho())
-2

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: (MR.Delta()^3*MR.E4()).order_at(infinity)
3
sage: MR.E2().order_at(infinity)
0
sage: (MR.J_inv()^2/MR.E4()).order_at(infinity)
-2
sage: el = MR((z^3-x*y)^2/(x^2*(x-y^2))).full_reduce()
sage: el
4*q - 304*q^2 + 8128*q^3 - 106144*q^4 + O(q^5)
sage: el.order_at(infinity)
1
sage: el.parent()
QuasiWeakModularForms(n=+Infinity, k=0, ep=1) over Integer Ring
sage: el.is_holomorphic()
False
sage: MR((z-x)^2+(x-y)^3).order_at(infinity)
2
sage: MR((x-y)^10).order_at(infinity)
10
sage: MR.zero().order_at(infinity)
+Infinity

sage: (MR.j_inv()*MR.f_i()^3).order_at(-1)
1
sage: (MR.j_inv()*MR.f_i()^3).order_at(i)
3
sage: (1/MR.f_inf()^2).order_at(-1)
0

sage: p = HyperbolicPlane().PD().get_point(I)
sage: MR((x-y)^10).order_at(p)
10
sage: MR.zero().order_at(p)
+Infinity

q_expansion(prec=None, fix_d=False, d_num_prec=None, fix_prec=False)

Returns the Fourier expansion of self.

INPUT:

• prec – An integer, the desired output precision O(q^prec).

  Default: None in which case the default precision of self.parent() 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.

• fix_prec – If fix_prec is not False (default)

  then the precision of the MFSeriesConstructor is increased such that the output has exactly the specified precision O(q^prec).

OUTPUT:

The Fourier expansion of self as a FormalPowerSeries or FormalLaurentSeries.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing, QuasiModularFormsRing
sage: j_inv = WeakModularFormsRing(red_hom=True).j_inv()
sage: j_inv.q_expansion(prec=3)
q^-1 + 31/(72*d) + 1823/(27648*d^2)*q + 10495/(2519424*d^3)*q^2 + O(q^3)

sage: E2 = QuasiModularFormsRing(n=5, red_hom=True).E2()
sage: E2.q_expansion(prec=3)
1 - 9/(200*d)*q - 369/(320000*d^2)*q^2 + O(q^3)
sage: E2.q_expansion(prec=3, fix_d=1)
1 - 9/200*q - 369/320000*q^2 + O(q^3)

sage: E6 = WeakModularFormsRing(n=5, red_hom=True).E6().full_reduce()
sage: Delta = WeakModularFormsRing(n=5, red_hom=True).Delta().full_reduce()
sage: E6.q_expansion(prec=3).prec() == 3
True
sage: (Delta/(E2^3-E6)).q_expansion(prec=3).prec() == 3
True
sage: (Delta/(E2^3-E6)^3).q_expansion(prec=3).prec() == 3
True
sage: ((E2^3-E6)/Delta^2).q_expansion(prec=3).prec() == 3
True
sage: ((E2^3-E6)^3/Delta).q_expansion(prec=3).prec() == 3
True

sage: x,y = var("x,y")
sage: el = WeakModularFormsRing()((x+1)/(x^3-y^2))
sage: el.q_expansion(prec=2, fix_prec = True)
2*d*q^-1 + O(1)
sage: el.q_expansion(prec=2)
2*d*q^-1 + 1/6 + 119/(41472*d)*q + O(q^2)

sage: j_inv = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()
sage: j_inv.q_expansion(prec=3)
q^-1 + 3/(8*d) + 69/(1024*d^2)*q + 1/(128*d^3)*q^2 + O(q^3)

sage: E2 = QuasiModularFormsRing(n=infinity, red_hom=True).E2()
sage: E2.q_expansion(prec=3)
1 - 1/(8*d)*q - 1/(512*d^2)*q^2 + O(q^3)
sage: E2.q_expansion(prec=3, fix_d=1)
1 - 1/8*q - 1/512*q^2 + O(q^3)

sage: E4 = WeakModularFormsRing(n=infinity, red_hom=True).E4().full_reduce()
sage: Delta = WeakModularFormsRing(n=infinity, red_hom=True).Delta().full_reduce()
sage: E4.q_expansion(prec=3).prec() == 3
True
sage: (Delta/(E2^2-E4)).q_expansion(prec=3).prec() == 3
True
sage: (Delta/(E2^2-E4)^3).q_expansion(prec=3).prec() == 3
True
sage: ((E2^2-E4)/Delta^2).q_expansion(prec=3).prec() == 3
True
sage: ((E2^2-E4)^3/Delta).q_expansion(prec=3).prec() == 3
True

sage: x,y = var("x,y")
sage: el = WeakModularFormsRing(n=infinity)((x+1)/(x-y^2))
sage: el.q_expansion(prec=2, fix_prec = True)
2*d*q^-1 + O(1)
sage: el.q_expansion(prec=2)
2*d*q^-1 + 1/2 + 39/(512*d)*q + O(q^2)

q_expansion_fixed_d(prec=None, d_num_prec=None, fix_prec=False)

Returns the Fourier expansion of self. The numerical (or exact) value for d is substituted.

INPUT:

• prec – An integer, the desired output precision O(q^prec).

  Default: None in which case the default precision of self.parent() is used.

• 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.

• fix_prec – If fix_prec is not False (default)

  then the precision of the MFSeriesConstructor is increased such that the output has exactly the specified precision O(q^prec).

OUTPUT:

The Fourier expansion of self as a FormalPowerSeries or FormalLaurentSeries.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing, QuasiModularFormsRing
sage: j_inv = WeakModularFormsRing(red_hom=True).j_inv()
sage: j_inv.q_expansion_fixed_d(prec=3)
q^-1 + 744 + 196884*q + 21493760*q^2 + O(q^3)

sage: E2 = QuasiModularFormsRing(n=5, red_hom=True).E2()
sage: E2.q_expansion_fixed_d(prec=3)
1.000000000000... - 6.380956565426...*q - 23.18584547617...*q^2 + O(q^3)

sage: x,y = var("x,y")
sage: WeakModularFormsRing()((x+1)/(x^3-y^2)).q_expansion_fixed_d(prec=2, fix_prec = True)
1/864*q^-1 + O(1)
sage: WeakModularFormsRing()((x+1)/(x^3-y^2)).q_expansion_fixed_d(prec=2)
1/864*q^-1 + 1/6 + 119/24*q + O(q^2)

sage: j_inv = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()
sage: j_inv.q_expansion_fixed_d(prec=3)
q^-1 + 24 + 276*q + 2048*q^2 + O(q^3)

sage: E2 = QuasiModularFormsRing(n=infinity, red_hom=True).E2()
sage: E2.q_expansion_fixed_d(prec=3)
1 - 8*q - 8*q^2 + O(q^3)

sage: x,y = var("x,y")
sage: WeakModularFormsRing(n=infinity)((x+1)/(x-y^2)).q_expansion_fixed_d(prec=2, fix_prec = True)
1/32*q^-1 + O(1)
sage: WeakModularFormsRing(n=infinity)((x+1)/(x-y^2)).q_expansion_fixed_d(prec=2)
1/32*q^-1 + 1/2 + 39/8*q + O(q^2)

sage: (WeakModularFormsRing(n=14).J_inv()^3).q_expansion_fixed_d(prec=2)
2.933373093...e-6*q^-3 + 0.0002320999814...*q^-2 + 0.009013529265...*q^-1 + 0.2292916854... + 4.303583833...*q + O(q^2)

q_expansion_vector(min_exp=None, max_exp=None, prec=None, **kwargs)

Return (part of) the Laurent series expansion of self as a vector.

INPUT:

• min_exp – An integer, specifying the first coefficient to be

  used for the vector. Default: None, meaning that the first non-trivial coefficient is used.

• max_exp – An integer, specifying the last coefficient to be

  used for the vector. Default: None, meaning that the default precision + 1 is used.

• prec – An integer, specifying the precision of the underlying

  Laurent series. Default: None, meaning that max_exp + 1 is used.

OUTPUT:

A vector of size max_exp - min_exp over the coefficient ring of self, determined by the corresponding Laurent series coefficients.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing
sage: f = WeakModularFormsRing(red_hom=True).j_inv()^3
sage: f.q_expansion(prec=3)
q^-3 + 31/(24*d)*q^-2 + 20845/(27648*d^2)*q^-1 + 7058345/(26873856*d^3) + 30098784355/(495338913792*d^4)*q + 175372747465/(17832200896512*d^5)*q^2 + O(q^3)
sage: v = f.q_expansion_vector(max_exp=1, prec=3)
sage: v
(1, 31/(24*d), 20845/(27648*d^2), 7058345/(26873856*d^3), 30098784355/(495338913792*d^4))
sage: v.parent()
Vector space of dimension 5 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: f.q_expansion_vector(min_exp=1, max_exp=2)
(30098784355/(495338913792*d^4), 175372747465/(17832200896512*d^5))
sage: f.q_expansion_vector(min_exp=1, max_exp=2, fix_d=True)
(541778118390, 151522053809760)

sage: f = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()^3
sage: f.q_expansion_fixed_d(prec=3)
q^-3 + 72*q^-2 + 2556*q^-1 + 59712 + 1033974*q + 14175648*q^2 + O(q^3)
sage: v = f.q_expansion_vector(max_exp=1, prec=3, fix_d=True)
sage: v
(1, 72, 2556, 59712, 1033974)
sage: v.parent()
Vector space of dimension 5 over Rational Field
sage: f.q_expansion_vector(min_exp=1, max_exp=2)
(516987/(8388608*d^4), 442989/(33554432*d^5))

rat()

Return the rational function representing self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
sage: ModularFormsRing(n=12).Delta().rat()
x^30*d - x^18*y^2*d

reduce(force=False)

In case self.parent().has_reduce_hom() == True (or force==True) and self is homogeneous the converted element lying in the corresponding homogeneous_part is returned.

Otherwise self is returned.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
sage: E2 = ModularFormsRing(n=7).E2().reduce()
sage: E2.parent()
QuasiModularFormsRing(n=7) over Integer Ring
sage: E2 = ModularFormsRing(n=7, red_hom=True).E2().reduce()
sage: E2.parent()
QuasiModularForms(n=7, k=2, ep=-1) over Integer Ring
sage: ModularFormsRing(n=7)(x+1).reduce().parent()
ModularFormsRing(n=7) over Integer Ring
sage: E2 = ModularFormsRing(n=7).E2().reduce(force=True)
sage: E2.parent()
QuasiModularForms(n=7, k=2, ep=-1) over Integer Ring
sage: ModularFormsRing(n=7)(x+1).reduce(force=True).parent()
ModularFormsRing(n=7) over Integer Ring

sage: y=var("y")
sage: ModularFormsRing(n=infinity)(x-y^2).reduce(force=True)
64*q - 512*q^2 + 1792*q^3 - 4096*q^4 + O(q^5)

reduced_parent()

Return the space with the analytic type of self. If self is homogeneous the corresponding FormsSpace is returned.

I.e. return the smallest known ambient space of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: Delta = QuasiMeromorphicModularFormsRing(n=7).Delta()
sage: Delta.parent()
QuasiMeromorphicModularFormsRing(n=7) over Integer Ring
sage: Delta.reduced_parent()
CuspForms(n=7, k=12, ep=1) over Integer Ring
sage: el = QuasiMeromorphicModularFormsRing()(x+1)
sage: el.parent()
QuasiMeromorphicModularFormsRing(n=3) over Integer Ring
sage: el.reduced_parent()
ModularFormsRing(n=3) over Integer Ring

sage: y=var("y")
sage: QuasiMeromorphicModularFormsRing(n=infinity)(x-y^2).reduced_parent()
ModularForms(n=+Infinity, k=4, ep=1) over Integer Ring
sage: QuasiMeromorphicModularFormsRing(n=infinity)(x*(x-y^2)).reduced_parent()
CuspForms(n=+Infinity, k=8, ep=1) over Integer Ring

serre_derivative()

Return the Serre derivative of self.

Note that the parent might (probably will) change. However a modular element is returned if self was already modular.

If parent.has_reduce_hom() == True then the result is reduced to be an element of the corresponding forms space if possible.

In particular this is the case if self is a (homogeneous) element of a forms space.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(n=7, red_hom=True)
sage: n = MR.hecke_n()
sage: Delta = MR.Delta().full_reduce()
sage: E2 = MR.E2().full_reduce()
sage: E4 = MR.E4().full_reduce()
sage: E6 = MR.E6().full_reduce()
sage: f_rho = MR.f_rho().full_reduce()
sage: f_i = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()

sage: f_rho.serre_derivative() == -1/n * f_i
True
sage: f_i.serre_derivative()   == -1/2 * E4 * f_rho
True
sage: f_inf.serre_derivative() == 0
True
sage: E2.serre_derivative()    == -(n-2)/(4*n) * (E2^2 + E4)
True
sage: E4.serre_derivative()    == -(n-2)/n * E6
True
sage: E6.serre_derivative()    == -1/2 * E4^2 - (n-3)/n * E6^2 / E4
True
sage: E6.serre_derivative().parent()
ModularForms(n=7, k=8, ep=1) over Integer Ring

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: Delta = MR.Delta().full_reduce()
sage: E2 = MR.E2().full_reduce()
sage: E4 = MR.E4().full_reduce()
sage: E6 = MR.E6().full_reduce()
sage: f_i = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()

sage: E4.serre_derivative()    == -E4 * f_i
True
sage: f_i.serre_derivative()   == -1/2 * E4
True
sage: f_inf.serre_derivative() == 0
True
sage: E2.serre_derivative()    == -1/4 * (E2^2 + E4)
True
sage: E4.serre_derivative()    == -E6
True
sage: E6.serre_derivative()    == -1/2 * E4^2 - E6^2 / E4
True
sage: E6.serre_derivative().parent()
ModularForms(n=+Infinity, k=8, ep=1) over Integer Ring

sqrt()

Return the square root of self if it exists.

I.e. the element corresponding to sqrt(self.rat()).

Whether this works or not depends on whether sqrt(self.rat()) works and coerces into self.parent().rat_field().

Note that the parent might (probably will) change.

If parent.has_reduce_hom() == True then the result is reduced to be an element of the corresponding forms space if possible.

In particular this is the case if self is a (homogeneous) element of a forms space.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: E2 = QuasiModularForms(k=2, ep=-1).E2()
sage: (E2^2).sqrt()
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 + O(q^5)
sage: (E2^2).sqrt() == E2
True

weight()

Return the weight of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: x,y,z,d = var("x,y,z,d")
sage: QuasiModularFormsRing()(x+y).weight() is None
True
sage: ModularForms(n=18).f_i().weight()
9/4
sage: ModularForms(n=infinity).f_inf().weight()
4