# Graded rings of modular forms#

This module contains functions to find generators for the graded ring of modular forms of given level.

AUTHORS:

• William Stein (2007-08-24): first version

• David Ayotte (2021-06): implemented category and Parent/Element frameworks

class sage.modular.modform.ring.ModularFormsRing(group, base_ring=Rational Field)#

Bases: Parent

The ring of modular forms (of weights 0 or at least 2) for a congruence subgroup of $$\SL_2(\ZZ)$$, with coefficients in a specified base ring.

EXAMPLES:

sage: ModularFormsRing(Gamma1(13))
Ring of Modular Forms for Congruence Subgroup Gamma1(13) over Rational Field
sage: m = ModularFormsRing(4); m
Ring of Modular Forms for Congruence Subgroup Gamma0(4) over Rational Field
sage: m.modular_forms_of_weight(2)
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(4) of weight 2 over Rational Field
sage: m.modular_forms_of_weight(10)
Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(4) of weight 10 over Rational Field
True
sage: m.generators()
[(2, 1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + O(q^10)),
(2, q + 4*q^3 + 6*q^5 + 8*q^7 + 13*q^9 + O(q^10))]
sage: m.q_expansion_basis(2,10)
[1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + O(q^10),
q + 4*q^3 + 6*q^5 + 8*q^7 + 13*q^9 + O(q^10)]
sage: m.q_expansion_basis(3,10)
[]
sage: m.q_expansion_basis(10,10)
[1 + 10560*q^6 + 3960*q^8 + O(q^10),
q - 8056*q^7 - 30855*q^9 + O(q^10),
q^2 - 796*q^6 - 8192*q^8 + O(q^10),
q^3 + 66*q^7 + 832*q^9 + O(q^10),
q^4 + 40*q^6 + 528*q^8 + O(q^10),
q^5 + 20*q^7 + 190*q^9 + O(q^10)]


Elements of modular forms ring can be initiated via multivariate polynomials (see from_polynomial()):

sage: M = ModularFormsRing(1)
sage: M.ngens()
2
sage: E4, E6 = polygens(QQ, 'E4, E6')
sage: M(E4)
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
sage: M(E6)
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
sage: M((E4^3 - E6^2)/1728)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)

Element#
change_ring(base_ring)#

Return the ring of modular forms over the given base ring and the same group as self.

INPUT:

• base_ring – a base ring, which should be $$\QQ$$, $$\ZZ$$, or the integers mod $$p$$ for some prime $$p$$.

EXAMPLES:

sage: M = ModularFormsRing(11); M
Ring of Modular Forms for Congruence Subgroup Gamma0(11) over Rational Field
sage: M.change_ring(Zmod(7))
Ring of Modular Forms for Congruence Subgroup Gamma0(11) over Ring of integers modulo 7
sage: M.change_ring(ZZ)
Ring of Modular Forms for Congruence Subgroup Gamma0(11) over Integer Ring

cuspidal_ideal_generators(maxweight=8, prec=None)#

Calculate generators for the ideal of cuspidal forms in this ring, as a module over the whole ring.

EXAMPLES:

sage: ModularFormsRing(Gamma0(3)).cuspidal_ideal_generators(maxweight=12)
[(6, q - 6*q^2 + 9*q^3 + 4*q^4 + O(q^5), q - 6*q^2 + 9*q^3 + 4*q^4 + 6*q^5 + O(q^6))]
sage: [k for k,f,F in ModularFormsRing(13, base_ring=ZZ).cuspidal_ideal_generators(maxweight=14)]
[4, 4, 4, 6, 6, 12]

cuspidal_submodule_q_expansion_basis(weight, prec=None)#

Calculate a basis of $$q$$-expansions for the space of cusp forms of weight weight for this group.

INPUT:

• weight (integer) – the weight

• prec (integer or None) – precision of $$q$$-expansions to return

ALGORITHM: Uses the method cuspidal_ideal_generators() to calculate generators of the ideal of cusp forms inside this ring. Then multiply these up to weight weight using the generators of the whole modular form space returned by q_expansion_basis().

EXAMPLES:

sage: R = ModularFormsRing(Gamma0(3))
sage: R.cuspidal_submodule_q_expansion_basis(20)
[q - 8532*q^6 - 88442*q^7 + O(q^8), q^2 + 207*q^6 + 24516*q^7 + O(q^8),
q^3 + 456*q^6 + O(q^8), q^4 - 135*q^6 - 926*q^7 + O(q^8), q^5 + 18*q^6 + 135*q^7 + O(q^8)]


We compute a basis of a space of very large weight, quickly (using this module) and slowly (using modular symbols), and verify that the answers are the same.

sage: A = R.cuspidal_submodule_q_expansion_basis(80, prec=30)  # long time (1s on sage.math, 2013)
sage: B = R.modular_forms_of_weight(80).cuspidal_submodule().q_expansion_basis(prec=30)  # long time (19s on sage.math, 2013)
sage: A == B # long time
True

from_polynomial(polynomial, gens=None)#

Convert the given polynomial to a graded form living in self. If gens is None then the list of generators given by the method gen_forms() will be used. Otherwise, gens should be a list of generators.

INPUT:

• polynomial – A multivariate polynomial. The variables names of the polynomial should be different from 'q'. The number of variable of this polynomial should equal the number of generators

• gens – list (default: None) of generators of the modular forms ring

OUTPUT: A GradedModularFormElement given by the polynomial relation polynomial.

EXAMPLES:

sage: M = ModularFormsRing(1)
sage: x,y = polygens(QQ, 'x,y')
sage: M.from_polynomial(x^2+y^3)
2 - 1032*q + 774072*q^2 - 77047584*q^3 - 11466304584*q^4 - 498052467504*q^5 + O(q^6)
sage: M = ModularFormsRing(Gamma0(6))
sage: M.ngens()
3
sage: x,y,z = polygens(QQ, 'x,y,z')
sage: M.from_polynomial(x+y+z)
1 + q + q^2 + 27*q^3 + q^4 + 6*q^5 + O(q^6)
sage: M.0 + M.1 + M.2
1 + q + q^2 + 27*q^3 + q^4 + 6*q^5 + O(q^6)
sage: P = x.parent()
sage: M.from_polynomial(P(1/2))
1/2


Note that the number of variables must be equal to the number of generators:

sage: x, y = polygens(QQ, 'x, y')
sage: M(x + y)
Traceback (most recent call last):
...
ValueError: the number of variables (2) must be equal to the number of generators of the modular forms ring (3)

gen(i)#

Return the $$i$$-th generator of self.

INPUT:

• i (Integer) – correspond to the $$i$$-th modular form generating the ring of modular forms.

OUTPUT: A GradedModularFormElement

EXAMPLES:

sage: M = ModularFormsRing(1)
sage: E4 = M.0; E4 # indirect doctest
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
sage: E6 = M.1; E6 # indirect doctest
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)

gen_forms(maxweight=8, start_gens=[], start_weight=2)#

Return a list of modular forms generating this ring (as an algebra over the appropriate base ring).

This method differs from generators() only in that it returns graded modular form objects, rather than bare $$q$$-expansions.

INPUT:

• maxweight (integer, default: 8) – calculate forms generating all forms up to this weight

• start_gens (list, default: []) – a list of modular forms. If this list is nonempty, we find a minimal generating set containing these forms

• start_weight (integer, default: 2) – calculate the graded subalgebra of forms of weight at least start_weight

Note

If called with the default values of start_gens (an empty list) and start_weight (2), the values will be cached for re-use on subsequent calls to this function. (This cache is shared with generators()). If called with non-default values for these parameters, caching will be disabled.

EXAMPLES:

sage: A = ModularFormsRing(Gamma0(11), Zmod(5)).gen_forms(); A
[1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + O(q^6),
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
q - 9*q^4 - 10*q^5 + O(q^6)]
sage: A.parent()
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

generators(maxweight=8, prec=10, start_gens=[], start_weight=2)#

If $$R$$ is the base ring of self, then this function calculates a set of modular forms which generate the $$R$$-algebra of all modular forms of weight up to maxweight with coefficients in $$R$$.

INPUT:

• maxweight (integer, default: 8) – check up to this weight for generators

• prec (integer, default: 10) – return $$q$$-expansions to this precision

• start_gens (list, default: []) – list of pairs $$(k, f)$$, or triples $$(k, f, F)$$, where:

• $$k$$ is an integer,

• $$f$$ is the $$q$$-expansion of a modular form of weight $$k$$, as a power series over the base ring of self,

• $$F$$ (if provided) is a modular form object corresponding to F.

If this list is nonempty, we find a minimal generating set containing these forms. If $$F$$ is not supplied, then $$f$$ needs to have sufficiently large precision (an error will be raised if this is not the case); otherwise, more terms will be calculated from the modular form object $$F$$.

• start_weight (integer, default: 2) – calculate the graded subalgebra of forms of weight at least start_weight.

OUTPUT:

a list of pairs (k, f), where f is the q-expansion to precision prec of a modular form of weight k.

gen_forms(), which does exactly the same thing, but returns Sage modular form objects rather than bare power series, and keeps track of a lifting to characteristic 0 when the base ring is a finite field.

Note

If called with the default values of start_gens (an empty list) and start_weight (2), the values will be cached for re-use on subsequent calls to this function. (This cache is shared with gen_forms()). If called with non-default values for these parameters, caching will be disabled.

EXAMPLES:

sage: ModularFormsRing(SL2Z).generators()
[(4, 1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + 60480*q^6 + 82560*q^7 + 140400*q^8 + 181680*q^9 + O(q^10)),
(6, 1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 - 4058208*q^6 - 8471232*q^7 - 17047800*q^8 - 29883672*q^9 + O(q^10))]
sage: s = ModularFormsRing(SL2Z).generators(maxweight=5, prec=3); s
[(4, 1 + 240*q + 2160*q^2 + O(q^3))]
sage: s.parent()
Power Series Ring in q over Rational Field

sage: ModularFormsRing(1).generators(prec=4)
[(4, 1 + 240*q + 2160*q^2 + 6720*q^3 + O(q^4)),
(6, 1 - 504*q - 16632*q^2 - 122976*q^3 + O(q^4))]
sage: ModularFormsRing(2).generators(prec=12)
[(2, 1 + 24*q + 24*q^2 + 96*q^3 + 24*q^4 + 144*q^5 + 96*q^6 + 192*q^7 + 24*q^8 + 312*q^9 + 144*q^10 + 288*q^11 + O(q^12)),
(4, 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + 30240*q^10 + O(q^12))]
sage: ModularFormsRing(4).generators(maxweight=2, prec=20)
[(2, 1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + 144*q^10 + 96*q^12 + 192*q^14 + 24*q^16 + 312*q^18 + O(q^20)),
(2, q + 4*q^3 + 6*q^5 + 8*q^7 + 13*q^9 + 12*q^11 + 14*q^13 + 24*q^15 + 18*q^17 + 20*q^19 + O(q^20))]


Here we see that for \Gamma_0(11) taking a basis of forms in weights 2 and 4 is enough to generate everything up to weight 12 (and probably everything else).:

sage: v = ModularFormsRing(11).generators(maxweight=12)
sage: len(v)
3
sage: [k for k, _ in v]
[2, 2, 4]
sage: from sage.modular.dims import dimension_modular_forms
sage: dimension_modular_forms(11,2)
2
sage: dimension_modular_forms(11,4)
4


For congruence subgroups not containing -1, we miss out some forms since we can’t calculate weight 1 forms at present, but we can still find generators for the ring of forms of weight $$\ge 2$$:

sage: ModularFormsRing(Gamma1(4)).generators(prec=10, maxweight=10)
[(2, 1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + O(q^10)),
(2, q + 4*q^3 + 6*q^5 + 8*q^7 + 13*q^9 + O(q^10)),
(3, 1 + 12*q^2 + 64*q^3 + 60*q^4 + 160*q^6 + 384*q^7 + 252*q^8 + O(q^10)),
(3, q + 4*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 32*q^6 + 48*q^7 + 64*q^8 + 73*q^9 + O(q^10))]


Using different base rings will change the generators:

sage: ModularFormsRing(Gamma0(13)).generators(maxweight=12, prec=4)
[(2, 1 + 2*q + 6*q^2 + 8*q^3 + O(q^4)),
(4, 1 + O(q^4)), (4, q + O(q^4)),
(4, q^2 + O(q^4)), (4, q^3 + O(q^4)),
(6, 1 + O(q^4)),
(6, q + O(q^4))]
sage: ModularFormsRing(Gamma0(13),base_ring=ZZ).generators(maxweight=12, prec=4)
[(2, 1 + 2*q + 6*q^2 + 8*q^3 + O(q^4)),
(4, q + 4*q^2 + 10*q^3 + O(q^4)),
(4, 2*q^2 + 5*q^3 + O(q^4)),
(4, q^2 + O(q^4)),
(4, -2*q^3 + O(q^4)),
(6, O(q^4)),
(6, O(q^4)),
(12, O(q^4))]
sage: [k for k,f in ModularFormsRing(1, QQ).generators(maxweight=12)]
[4, 6]
sage: [k for k,f in ModularFormsRing(1, ZZ).generators(maxweight=12)]
[4, 6, 12]
sage: [k for k,f in ModularFormsRing(1, Zmod(5)).generators(maxweight=12)]
[4, 6]
sage: [k for k,f in ModularFormsRing(1, Zmod(2)).generators(maxweight=12)]
[4, 6, 12]


An example where start_gens are specified:

sage: M = ModularForms(11, 2); f = (M.0 + M.1).qexp(8)
sage: ModularFormsRing(11).generators(start_gens = [(2, f)])
Traceback (most recent call last):
...
ValueError: Requested precision cannot be higher than precision of approximate starting generators!
sage: f = (M.0 + M.1).qexp(10); f
1 + 17/5*q + 26/5*q^2 + 43/5*q^3 + 94/5*q^4 + 77/5*q^5 + 154/5*q^6 + 86/5*q^7 + 36*q^8 + 146/5*q^9 + O(q^10)
sage: ModularFormsRing(11).generators(start_gens = [(2, f)])
[(2, 1 + 17/5*q + 26/5*q^2 + 43/5*q^3 + 94/5*q^4 + 77/5*q^5 + 154/5*q^6 + 86/5*q^7 + 36*q^8 + 146/5*q^9 + O(q^10)),
(2, 1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + 24*q^6 + 24*q^7 + 36*q^8 + 36*q^9 + O(q^10)),
(4, 1 + O(q^10))]

gens(maxweight=8, start_gens=[], start_weight=2)#

Return a list of modular forms generating this ring (as an algebra over the appropriate base ring).

This method differs from generators() only in that it returns graded modular form objects, rather than bare $$q$$-expansions.

INPUT:

• maxweight (integer, default: 8) – calculate forms generating all forms up to this weight

• start_gens (list, default: []) – a list of modular forms. If this list is nonempty, we find a minimal generating set containing these forms

• start_weight (integer, default: 2) – calculate the graded subalgebra of forms of weight at least start_weight

Note

If called with the default values of start_gens (an empty list) and start_weight (2), the values will be cached for re-use on subsequent calls to this function. (This cache is shared with generators()). If called with non-default values for these parameters, caching will be disabled.

EXAMPLES:

sage: A = ModularFormsRing(Gamma0(11), Zmod(5)).gen_forms(); A
[1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + O(q^6),
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
q - 9*q^4 - 10*q^5 + O(q^6)]
sage: A.parent()
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

group()#

Return the congruence subgroup for which this is the ring of modular forms.

EXAMPLES:

sage: R = ModularFormsRing(Gamma1(13))
sage: R.group() is Gamma1(13)
True

modular_forms_of_weight(weight)#

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

EXAMPLES:

sage: R = ModularFormsRing(13)
sage: R.modular_forms_of_weight(10)
Modular Forms space of dimension 11 for Congruence Subgroup Gamma0(13) of weight 10 over Rational Field
sage: ModularFormsRing(Gamma1(13)).modular_forms_of_weight(3)
Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(13) of weight 3 over Rational Field

ngens()#

Return the number of generators of self

EXAMPLES:

sage: ModularFormsRing(1).ngens()
2
sage: ModularFormsRing(Gamma0(2)).ngens()
2
sage: ModularFormsRing(Gamma1(13)).ngens() # long time
33


Warning

Computing the number of generators of a graded ring of modular form for a certain congruence subgroup can be very long.

one()#

Return the one element of this ring.

EXAMPLES:

sage: M = ModularFormsRing(1)
sage: u = M.one(); u
1
sage: u.is_one()
True
sage: u + u
2
sage: E4 = ModularForms(1,4).0; E4
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
sage: E4 * u
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)

polynomial_ring(names, gens=None)#

Return a polynomial ring of which self is a quotient.

INPUT:

• names – a list or tuple of names (strings), or a comma separated string

• gens (default: None) – (list) a list of generator of self. If gens is None then the generators returned by gen_forms() is used instead.

OUTPUT: A multivariate polynomial ring in the variable names. Each variable of the polynomial ring correspond to a generator given in gens (following the ordering of the list).

EXAMPLES:

sage: M = ModularFormsRing(1)
sage: gens = M.gen_forms()
sage: M.polynomial_ring('E4, E6', gens)
Multivariate Polynomial Ring in E4, E6 over Rational Field
sage: M = ModularFormsRing(Gamma0(8))
sage: gens = M.gen_forms()
sage: M.polynomial_ring('g', gens)
Multivariate Polynomial Ring in g0, g1, g2 over Rational Field


The degrees of the variables are the weights of the corresponding forms:

sage: M = ModularFormsRing(1)
sage: P.<E4, E6> = M.polynomial_ring()
sage: E4.degree()
4
sage: E6.degree()
6
sage: (E4*E6).degree()
10

q_expansion_basis(weight, prec=None, use_random=True)#

Calculate a basis of q-expansions for the space of modular forms of the given weight for this group, calculated using the ring generators given by find_generators.

INPUT:

• weight (integer) – the weight

• prec (integer or None, default: None) – power series precision. If None, the precision defaults to the Sturm bound for the requested level and weight.

• use_random (boolean, default: True) – whether or not to use a randomized algorithm when building up the space of forms at the given weight from known generators of small weight.

EXAMPLES:

sage: m = ModularFormsRing(Gamma0(4))
sage: m.q_expansion_basis(2,10)
[1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + O(q^10),
q + 4*q^3 + 6*q^5 + 8*q^7 + 13*q^9 + O(q^10)]
sage: m.q_expansion_basis(3,10)
[]

sage: X = ModularFormsRing(SL2Z)
sage: X.q_expansion_basis(12, 10)
[1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + 34417656000*q^6 + 187489935360*q^7 + 814879774800*q^8 + 2975551488000*q^9 + O(q^10),
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 + O(q^10)]


We calculate a basis of a massive modular forms space, in two ways. Using this module is about twice as fast as Sage’s generic code.

sage: A = ModularFormsRing(11).q_expansion_basis(30, prec=40) # long time (5s)
sage: B = ModularForms(Gamma0(11), 30).q_echelon_basis(prec=40) # long time (9s)
sage: A == B # long time
True


Check that absurdly small values of prec don’t mess things up:

sage: ModularFormsRing(11).q_expansion_basis(10, prec=5)
[1 + O(q^5), q + O(q^5), q^2 + O(q^5), q^3 + O(q^5),
q^4 + O(q^5), O(q^5), O(q^5), O(q^5), O(q^5), O(q^5)]

some_elements()#

Return a list of generators of self.

EXAMPLES:

sage: ModularFormsRing(1).some_elements()
[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)]

zero()#

Return the zero element of this ring.

EXAMPLES:

sage: M = ModularFormsRing(1)
sage: zer = M.zero(); zer
0
sage: zer.is_zero()
True
sage: E4 = ModularForms(1,4).0; E4
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
sage: E4 + zer
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
sage: zer * E4
0
sage: E4 * zer
0