Graded quasimodular forms ring

Let $E_2$ be the weight 2 Eisenstein series defined by

$$E_2(z)=1-24\sum _{n=1}^{\mathrm{\infty }}\sigma (n)q^n$$

where $\sigma$ is the sum of divisors function and $q=\exp (2\pi iz)$ is the classical parameter at infinity, with $\mathrm{im}(z)>0$. This weight 2 Eisenstein series is not a modular form as it does not satisfy the modularity condition:

$$z^2{E}_2(-1/z)=E_2(z)+\frac{6}{\pi iz}.$$

$E_2$ is a quasimodular form of weight 2. General quasimodular forms of given weight can also be defined. We denote by $QM$ the graded ring of quasimodular forms for the full modular group ${\mathrm{SL}}_2(\mathbf{Z})$.

The SageMath implementation of the graded ring of quasimodular forms uses the following isomorphism:

$$QM\cong M_{\ast }[E_2]$$

where $M_{\ast }\cong \mathbf{C}[E_4,E_6]$ is the graded ring of modular forms for ${\mathrm{SL}}_2(\mathbf{Z})$. (see sage.modular.modform.ring.ModularFormsRing).

More generally, if $\mathrm{\Gamma }\le {\mathrm{SL}}_2(\mathbf{Z})$ is a congruence subgroup, then the graded ring of quasimodular forms for $\mathrm{\Gamma }$ is given by $M_{\ast }(\mathrm{\Gamma })[E_2]$ where $M_{\ast }(\mathrm{\Gamma })$ is the ring of modular forms for $\mathrm{\Gamma }$.

The SageMath implementation of the graded quasimodular forms ring allows computation of a set of generators and perform usual arithmetic operations.

EXAMPLES:

sage: QM = QuasiModularForms(1); QM
Ring of Quasimodular Forms for Modular Group SL(2,Z) over Rational Field
sage: QM.category()
Category of commutative graded algebras over Rational Field
sage: QM.gens()
(1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6),
 1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6),
 1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6))
sage: E2 = QM.0; E4 = QM.1; E6 = QM.2
sage: E2 * E4 + E6
2 - 288*q - 20304*q^2 - 185472*q^3 - 855216*q^4 - 2697408*q^5 + O(q^6)
sage: E2.parent()
Ring of Quasimodular Forms for Modular Group SL(2,Z) over Rational Field

The polygen method also return the weight-2 Eisenstein series as a polynomial variable over the ring of modular forms:

sage: QM = QuasiModularForms(1)
sage: E2 = QM.polygen(); E2
E2
sage: E2.parent()
Univariate Polynomial Ring in E2 over Ring of Modular Forms for Modular Group SL(2,Z) over Rational Field

An element of a ring of quasimodular forms can be created via a list of modular forms or graded modular forms. The $i$-th index of the list will correspond to the $i$-th coefficient of the polynomial in $E_2$:

sage: QM = QuasiModularForms(1)
sage: E2 = QM.0
sage: Delta = CuspForms(1, 12).0
sage: E4 = ModularForms(1, 4).0
sage: F = QM([Delta, E4, Delta + E4]); F
2 + 410*q - 12696*q^2 - 50424*q^3 + 1076264*q^4 + 10431996*q^5 + O(q^6)
sage: F == Delta + E4 * E2 + (Delta + E4) * E2^2
True

One may also create rings of quasimodular forms for certain congruence subgroups:

sage: QM = QuasiModularForms(Gamma0(5)); QM
Ring of Quasimodular Forms for Congruence Subgroup Gamma0(5) over Rational Field
sage: QM.ngens()
4

The first generator is the weight 2 Eisenstein series:

sage: E2 = QM.0; E2
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)

The other generators correspond to the generators given by the method sage.modular.modform.ring.ModularFormsRing.gens():

sage: QM.gens()
(1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6),
 1 + 6*q + 18*q^2 + 24*q^3 + 42*q^4 + 6*q^5 + O(q^6),
 1 + 240*q^5 + O(q^6),
 q + 10*q^3 + 28*q^4 + 35*q^5 + O(q^6))
sage: QM.modular_forms_subring().gens()
[1 + 6*q + 18*q^2 + 24*q^3 + 42*q^4 + 6*q^5 + O(q^6),
 1 + 240*q^5 + O(q^6),
 q + 10*q^3 + 28*q^4 + 35*q^5 + O(q^6)]

It is possible to convert a graded quasimodular form into a polynomial where each variable corresponds to a generator of the ring:

sage: QM = QuasiModularForms(1)
sage: E2, E4, E6 = QM.gens()
sage: F = E2*E4*E6 + E6^2; F
2 - 1296*q + 91584*q^2 + 14591808*q^3 + 464670432*q^4 + 6160281120*q^5 + O(q^6)
sage: p = F.polynomial('E2, E4, E6'); p
E2*E4*E6 + E6^2
sage: P = p.parent(); P
Multivariate Polynomial Ring in E2, E4, E6 over Rational Field

The generators of the polynomial ring have degree equal to the weight of the corresponding form:

sage: P.inject_variables()
Defining E2, E4, E6
sage: E2.degree()
2
sage: E4.degree()
4
sage: E6.degree()
6

This works also for congruence subgroup:

sage: QM = QuasiModularForms(Gamma1(4))
sage: QM.ngens()
5
sage: QM.polynomial_ring()
Multivariate Polynomial Ring in E2, E2_0, E2_1, E3_0, E3_1 over Rational Field
sage: (QM.0 + QM.1*QM.0^2 + QM.3 + QM.4^3).polynomial()
E3_1^3 + E2^2*E2_0 + E3_0 + E2

One can also convert a multivariate polynomial into a quasimodular form:

sage: QM.polynomial_ring().inject_variables()
Defining E2, E2_0, E2_1, E3_0, E3_1
sage: QM.from_polynomial(E3_1^3 + E2^2*E2_0 + E3_0 + E2)
3 - 72*q + 396*q^2 + 2081*q^3 + 19752*q^4 + 98712*q^5 + O(q^6)

Note

• Currently, the only supported base ring is the Rational Field;

• Spaces of quasimodular forms of fixed weight are not yet implemented.

REFERENCE:

See section 5.3 (page 58) of [Zag2008]

AUTHORS:

• David Ayotte (2021-03-18): initial version

class sage.modular.quasimodform.ring.QuasiModularForms(group=1, base_ring=Rational Field, name='E2')

Bases: Parent, UniqueRepresentation

The graded ring of quasimodular forms for the full modular group ${\mathrm{SL}}_2(\mathbf{Z})$, with coefficients in a ring.

EXAMPLES:

sage: QM = QuasiModularForms(1); QM
Ring of Quasimodular Forms for Modular Group SL(2,Z) over Rational Field
sage: QM.gens()
(1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6),
 1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6),
 1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6))

It is possible to access the weight 2 Eisenstein series:

sage: QM.weight_2_eisenstein_series()
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)

Currently, the only supported base ring is the rational numbers:

sage: QuasiModularForms(1, GF(5))
Traceback (most recent call last):
...
NotImplementedError: base ring other than Q are not yet supported for quasimodular forms ring

Element

alias of QuasiModularFormsElement

basis_of_weight(weight)

Return a basis of elements generating the subspace of the given weight.

INPUT:

• weight – integer; the weight of the subspace

OUTPUT: list of quasimodular forms of the given weight

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: QM.basis_of_weight(12)
[q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6),
 1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + 3199218815520/691*q^5 + O(q^6),
 1 - 288*q - 129168*q^2 - 1927296*q^3 + 65152656*q^4 + 1535768640*q^5 + O(q^6),
 1 + 432*q + 39312*q^2 - 1711296*q^3 - 14159664*q^4 + 317412000*q^5 + O(q^6),
 1 - 576*q + 21168*q^2 + 308736*q^3 - 15034608*q^4 - 39208320*q^5 + O(q^6),
 1 + 144*q - 17712*q^2 + 524736*q^3 - 2279088*q^4 - 79760160*q^5 + O(q^6),
 1 - 144*q + 8208*q^2 - 225216*q^3 + 2634192*q^4 + 1488672*q^5 + O(q^6)]
sage: QM = QuasiModularForms(Gamma1(3))
sage: QM.basis_of_weight(3)
[1 + 54*q^2 + 72*q^3 + 432*q^5 + O(q^6),
 q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + O(q^6)]
sage: QM.basis_of_weight(5)
[1 - 90*q^2 - 240*q^3 - 3744*q^5 + O(q^6),
 q + 15*q^2 + 81*q^3 + 241*q^4 + 624*q^5 + O(q^6),
 1 - 24*q - 18*q^2 - 1320*q^3 - 5784*q^4 - 10080*q^5 + O(q^6),
 q - 21*q^2 - 135*q^3 - 515*q^4 - 1392*q^5 + O(q^6)]

from_polynomial(polynomial)

Convert the given polynomial $P(x,\dots ,y)$ to the graded quasiform $P(g_0,\dots ,g_n)$ where the $g_i$ are the generators given by gens().

INPUT:

• polynomial – a multivariate polynomial

OUTPUT: the graded quasimodular forms $P(g_0,\dots ,g_n)$

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: P.<x, y, z> = QQ[]
sage: QM.from_polynomial(x)
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)
sage: QM.from_polynomial(x) == QM.0
True
sage: QM.from_polynomial(y) == QM.1
True
sage: QM.from_polynomial(z) == QM.2
True
sage: QM.from_polynomial(x^2 + y + x*z + 1)
4 - 336*q - 2016*q^2 + 322368*q^3 + 3691392*q^4 + 21797280*q^5 + O(q^6)
sage: QM = QuasiModularForms(Gamma0(2))
sage: P = QM.polynomial_ring()
sage: P.inject_variables()
Defining E2, E2_0, E4_0
sage: QM.from_polynomial(E2)
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)
sage: QM.from_polynomial(E2 + E4_0*E2_0) == QM.0 + QM.2*QM.1
True

Naturally, the number of variable must not exceed the number of generators:

sage: P = PolynomialRing(QQ, 'F', 4)
sage: P.inject_variables()
Defining F0, F1, F2, F3
sage: QM.from_polynomial(F0 + F1 + F2 + F3)
Traceback (most recent call last):
...
ValueError: the number of variables (4) of the given polynomial cannot exceed the number of generators (3) of the quasimodular forms ring

gen(n)

Return the $n$-th generator of the quasimodular forms ring.

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: QM.0
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)
sage: QM.1
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
sage: QM.2
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
sage: QM = QuasiModularForms(5)
sage: QM.0
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)
sage: QM.1
1 + 6*q + 18*q^2 + 24*q^3 + 42*q^4 + 6*q^5 + O(q^6)
sage: QM.2
1 + 240*q^5 + O(q^6)
sage: QM.3
q + 10*q^3 + 28*q^4 + 35*q^5 + O(q^6)
sage: QM.4
Traceback (most recent call last):
...
IndexError: tuple index out of range

generators()

alias of gens().

gens()

Return a tuple of generators of the quasimodular forms ring.

Note that the generators of the modular forms subring are the one given by the method sage.modular.modform.ring.ModularFormsRing.gen_forms()

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: QM.gens()
(1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6),
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6),
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6))
sage: QM.modular_forms_subring().gen_forms()
[1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6),
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)]
sage: QM = QuasiModularForms(5)
sage: QM.gens()
(1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6),
1 + 6*q + 18*q^2 + 24*q^3 + 42*q^4 + 6*q^5 + O(q^6),
1 + 240*q^5 + O(q^6),
q + 10*q^3 + 28*q^4 + 35*q^5 + O(q^6))

An alias of this method is generators:

sage: QuasiModularForms(1).generators()
(1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6),
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6),
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6))

group()

Return the congruence subgroup attached to the given quasimodular forms ring.

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: QM.group()
Modular Group SL(2,Z)
sage: QM.group() is SL2Z
True
sage: QuasiModularForms(3).group()
Congruence Subgroup Gamma0(3)
sage: QuasiModularForms(Gamma1(5)).group()
Congruence Subgroup Gamma1(5)

modular_forms_of_weight(weight)

Return the space of modular forms on this group of the given weight.

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: QM.modular_forms_of_weight(12)
Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
sage: QM = QuasiModularForms(Gamma1(3))
sage: QM.modular_forms_of_weight(4)
Modular Forms space of dimension 2 for Congruence Subgroup Gamma1(3) of weight 4 over Rational Field

modular_forms_subring()

Return the subring of modular forms of this ring of quasimodular forms.

EXAMPLES:

sage: QuasiModularForms(1).modular_forms_subring()
Ring of Modular Forms for Modular Group SL(2,Z) over Rational Field
sage: QuasiModularForms(5).modular_forms_subring()
Ring of Modular Forms for Congruence Subgroup Gamma0(5) over Rational Field

ngens()

Return the number of generators of the given graded quasimodular forms ring.

EXAMPLES:

sage: QuasiModularForms(1).ngens()
3

one()

Return the one element of this ring.

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: QM.one()
1
sage: QM.one().is_one()
True

polygen()

Return the generator of this quasimodular form space as a polynomial ring over the modular form subring.

Note that this generator correspond to the weight-2 Eisenstein series. The default name of this generator is E2.

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: QM.polygen()
E2
sage: QuasiModularForms(1, name='X').polygen()
X
sage: QM.polygen().parent()
Univariate Polynomial Ring in E2 over Ring of Modular Forms for Modular Group SL(2,Z) over Rational Field

polynomial_ring(names=None)

Return a multivariate polynomial ring of which the quasimodular forms ring is a quotient.

In the case of the full modular group, this ring is $R[E_2,E_4,E_6]$ where $E_2$, $E_4$ and $E_6$ have degrees 2, 4 and 6 respectively.

INPUT:

• names– string (default: None); list or tuple of names (strings), or a comma separated string. Defines the names for the generators of the multivariate polynomial ring. The default names are of the following form:

  • E2 denotes the weight 2 Eisenstein series;

  • Ek_i and Sk_i denote the $i$-th basis element of the weight $k$ Eisenstein subspace and cuspidal subspace respectively;

  • If the level is one, the default names are E2, E4 and E6;

  • In any other cases, we use the letters Fk, Gk, Hk, …, FFk, FGk, … to denote any generator of weight $k$.

OUTPUT: a multivariate polynomial ring in the variables names

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: P = QM.polynomial_ring(); P
Multivariate Polynomial Ring in E2, E4, E6 over Rational Field
sage: P.inject_variables()
Defining E2, E4, E6
sage: E2.degree()
2
sage: E4.degree()
4
sage: E6.degree()
6

Example when the level is not one:

sage: QM = QuasiModularForms(Gamma0(29))
sage: P_29 = QM.polynomial_ring()
sage: P_29
Multivariate Polynomial Ring in E2, F2, S2_0, S2_1, E4_0, F4, G4, H4 over Rational Field
sage: P_29.inject_variables()
Defining E2, F2, S2_0, S2_1, E4_0, F4, G4, H4
sage: F2.degree()
2
sage: E4_0.degree()
4

The name Sk_i stands for the $i$-th basis element of the cuspidal subspace of weight $k$:

sage: F2 = QM.from_polynomial(S2_0)
sage: F2.qexp(10)
q - q^4 - q^5 - q^6 + 2*q^7 - 2*q^8 - 2*q^9 + O(q^10)
sage: CuspForms(Gamma0(29), 2).0.qexp(10)
q - q^4 - q^5 - q^6 + 2*q^7 - 2*q^8 - 2*q^9 + O(q^10)
sage: F2 == CuspForms(Gamma0(29), 2).0
True

The name Ek_i stands for the $i$-th basis element of the Eisenstein subspace of weight $k$:

sage: F4 = QM.from_polynomial(E4_0)
sage: F4.qexp(30)
1 + 240*q^29 + O(q^30)
sage: EisensteinForms(Gamma0(29), 4).0.qexp(30)
1 + 240*q^29 + O(q^30)
sage: F4 == EisensteinForms(Gamma0(29), 4).0
True

One may also choose the name of the variables:

sage: QM = QuasiModularForms(1)
sage: QM.polynomial_ring(names="P, Q, R")
Multivariate Polynomial Ring in P, Q, R over Rational Field

quasimodular_forms_of_weight(weight)

Return the space of quasimodular forms on this group of the given weight.

INPUT:

• weight – integer

OUTPUT: a quasimodular forms space of the given weight

EXAMPLES:

sage: QuasiModularForms(1).quasimodular_forms_of_weight(4)
Traceback (most recent call last):
...
NotImplementedError: spaces of quasimodular forms of fixed weight not yet implemented

some_elements()

Return a list of generators of self.

EXAMPLES:

sage: QuasiModularForms(1).some_elements()
(1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6),
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6),
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6))

weight_2_eisenstein_series()

Return the weight 2 Eisenstein series.

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: E2 = QM.weight_2_eisenstein_series(); E2
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)
sage: E2.parent()
Ring of Quasimodular Forms for Modular Group SL(2,Z) over Rational Field

zero()

Return the zero element of this ring.

EXAMPLES:

sage: QM = QuasiModularForms(1)
sage: QM.zero()
0
sage: QM.zero().is_zero()
True