Bases

Any space of modular forms that can be created in Magma is of the form \(M_{\mathbb{Z}}{\otimes}_{{\mathbb{Z}}} R\) for some ring \(R\) and some space \(M_{\mathbb{Z}}\) of modular forms defined over \({\mathbb{Z}}\). The basis of \(M\) is the image in \(M\) of Basis(M_Z), and Basis(M_Z) is in Hermite normal form.

Note that in order to determine this basis over \({\mathbb{Z}}\), it is necessary to compute \(q\)-expansions up to a “Sturm bound” for \(M\) (see PrecisionBound). The precision used internally will therefore be at least as large as this bound. If desired, however, one can compute q-expansions to lower precision by working directly with spaces of modular symbols.

Basis(M): ModFrm -> SeqEnum

The canonical basis of the space of modular forms or half-integral weight forms \(M\).

Basis(M, prec): ModFrm, RngIntElt -> [RngSerPowElt]

qExpansionBasis(M, prec): ModFrm, RngIntElt -> [RngSerPowElt]

A sequence containing q-expansions (to the specified precision) of the elements of Basis(M).

PrecisionBound(M : parameters): ModFrm -> RngIntElt

Exact: BoolElt                    Default: false

An integer \(b\) such that \(f + O(q^b)\) determines any modular form \(f\) in the given space \(M\) of modular forms or half-integral weight forms. If the optional parameter Exact is set to true, or if a q-expansion basis has already been computed for \(M\), then the result is best-possible, ie the smallest integer \(b\) such that \(f+O(q^b)\) determines any modular form \(f\) in \(M\). Otherwise it is a “Sturm bound” similar to (although in some cases sharper than) the bounds given in section 9.4 of [Stein, 2007].

Note: In some much older versions of Magma the default was Exact := true.

RModule(M): ModFrm -> ModTupRng, Map

RSpace(M): ModFrm -> ModTupRng, Map

VectorSpace(M): ModFrm -> ModTupRng, Map

Ring: Rng                    Default: 

An abstract free module isomorphic to the given space of modular forms M, over the same base ring (unless Ring is specified). The second returned object is a map from the abstract module to/from M.

This function is needed when one wants to use linear algebra functions on \(M\) (since in Magma, a space of modular forms is not a subtype of vector space).

Example: Bases (ex-fd9bbd)

> M := ModularForms(Gamma1(16),3); M;
Space of modular forms on Gamma_1(16) of weight 3 and dimension 23
over Integer Ring.
> Dimension(CuspidalSubspace(M));
9
> SetPrecision(M,19);
> Basis(NewSubspace(CuspidalSubspace(M)))[1];
q - 76*q^8 + 39*q^9 + 132*q^10 - 44*q^11 + 84*q^12 - 144*q^13 -
232*q^14 + 120*q^15 + 160*q^16 + 158*q^17 - 76*q^18 + O(q^19)

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We can print the whole basis to less precision:

> SetPrecision(M,10);
> Basis(NewSubspace(CuspidalSubspace(M)));
[
    q - 76*q^8 + 39*q^9 + O(q^10),
    q^2 + q^7 - 58*q^8 + 30*q^9 + O(q^10),
    q^3 + 2*q^7 - 42*q^8 + 18*q^9 + O(q^10),
    q^4 + q^7 - 26*q^8 + 13*q^9 + O(q^10),
    q^5 + 2*q^7 - 18*q^8 + 5*q^9 + O(q^10),
    q^6 + 2*q^7 - 12*q^8 + 3*q^9 + O(q^10),
    3*q^7 - 8*q^8 + O(q^10)
]

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Note the coefficient \(3\) of \(q^7\), which emphasizes that this is not the reduced echelon form over a field, but the image of a reduced form over the integers:

> MQ := BaseExtend(M,RationalField());
> Basis(NewSubspace(CuspidalSubspace(MQ)))[7];
3*q^7 - 8*q^8 + O(q^10)

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