The following intrinsics give the q-expansion of a modular form (about the cusp Infinity).
The q-expansion of the modular form f. This is an element of the power series ring over the base ring of the parent of f.
The q-expansion of the modular form f to absolute precision prec. This is an element of the power series ring over the base ring of the parent of f.
In the context of modular forms, PowerSeries is a synonym for qExpansion.
The default printing precision for elements of the space M of modular forms. This is the precision that is used when printing elements of M if the user does not specifically make a choice. The hard-coded default value is 8.
Set the default printing precision for elements of the space M of modular forms.
Exact: BoolElt Default: true
The smallest integer b such that f + O(q^b) determines any modular form f in M. If the optional parameter Exact is set to false, then this intrinsic returns some integer b such that f + O(q^b) determines any modular form f in M, but b is probably not minimal.
> M := ModularForms(Gamma1(11),3); M;
Space of modular forms on Gamma_1(11) of weight 3 and dimension 15
over Integer Ring.
> f := M.1;
> f;
1 + O(q^8)
> qExpansion(f);
1 + O(q^8)
> qExpansion(f,17);
1 + 763774*q^15 - 5457936*q^16 + O(q^17)
> PowerSeries(f,20); // same as qExpansion(f,20)
1 + 763774*q^15 - 5457936*q^16 + 14709156*q^17 - 12391258*q^18 -
21614340*q^19 + O(q^20)
The "big-oh" notation is supported via addition of a modular
form and a power series.
> M<q> := Parent(f); > Parent(q); Power series ring in q over Integer Ring > f + O(q^17); 1 + 763774*q^15 - 5457936*q^16 + O(q^17) > 5*q - O(q^17) + f; 1 + 5*q + 763774*q^15 - 5457936*q^16 + O(q^17) > 5*q + f; 1 + 5*q + O(q^8)Default printing precision can be set using the command SetPrecision.
> SetPrecision(M,16); > f; 1 + 763774*q^15 + O(q^16)
> function InftyIsWP(N) > S := CuspidalSubspace(ModularForms(Gamma0(N),2)); > assert Dimension(S) ge 2; > return (PrecisionBound(S)-1) ne Dimension(S); > end function; > [<N,InftyIsWP(N)> : N in [97..100]]; [ <97, false>, <98, true>, <99, false>, <100, true> ]It is an open problem to give a simple characterization of the integers N such that Infinity is a Weierstrass point on X_0(N), though Atkin and others have made significant progress on this problem (see, e.g., 1967 Annals paper [Atk67]). I verified that if N<3223 is square free, then Infinity is not a Weierstrass point on X_0(N), which suggests a nice conjecture.