q-Expansions

The following intrinsics give the q-expansion of a modular form (about the cusp ∞).

Note that q-expansions are printed by default only to precision O(q¹2). This may be adjusted using SetPrecision (see below), which should be used to control printing only; to control the amount of precision computed internally, instead use qExpansion or qExpansionBasis and specify the desired precision.

qExpansion(f) : ModFrmElt -> RngSerPowElt

qExpansion(f, prec) : ModFrmElt, RngIntElt -> RngSerPowElt

PowerSeries(f) : ModFrmElt -> RngSerPowElt

PowerSeries(f, prec) : ModFrmElt, RngIntElt -> RngSerPowElt

    GaloisOrbitRep: BoolElt             Default: false

The q-expansion (at the cusp ∞) of the modular form (or half-integral weight form) f to absolute precision prec. This is an element of the power series ring over the base ring of the parent of f.

If GaloisOrbitRep is set to true, then returns its Galois orbit representative, with a different variable name to mark the conjugate form.

Coefficient(f, n) : ModFrmElt, RngIntElt -> RngElt

The nth coefficient of the q-expansion of the modular form f.

Precision(M) : ModFrm -> RngIntElt

SetPrecision(M, prec) : ModFrm, RngIntElt ->

When an element of the space M is printed, the q-expansion is displayed to this precision. The default value is 12.

Important note: This controls only printing. It does not control the precision used during calculations. For instance, the precision to which q-expansions are computed is controlled by the second argument in qExpansion and qExpansionBasis.

> 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)
> Coefficient(f,16);  // f is a modular form, so has infinite precision
-5457936
> 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)

> 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)

> 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 : Exact := true)-1) ne Dimension(S);
> end function;
> [<N,InftyIsWP(N)> : N in [97..100]];
[ <97, false>, <98, true>, <99, false>, <100, true> ]