The verbosity level for modular symbols computations can be set using the command SetVerbose("ModularSymbols",n), where n is 0 (silent), 1 (verbose), or 2 (very verbose).
Spaces of modular symbols belong to the category ModSym, and groups of Dirichlet characters form a category GrpDrch. The category SetCsp has exactly one object Cusps(), which is the set PP^1(Q) = Qunion {Infinity} introduced above. The element Infinity of PP^1(Q) is entered using the expression Cusps()!Infinity().
> M := ModularSymbols(11,2); M;
Full Modular symbols space of level 11, weight 2, and dimension 3
> Type(M);
ModSym
> Basis(M);
[
{-1/7, 0},
{-1/5, 0},
{oo, 0}
]
> M!<1,[1/5,1]>;
{-1/5, 0}
> // the modular symbols {1/5,1} and {-1/5,0} are equal.
> Type(M!<1,[1/5,1]>);
ModSymElt
Using SetVerbose, we can see how the computation progresses.
> SetVerbose("ModularSymbols",2);
> M := ModularSymbols(11,2);
Computing space of modular symbols of level 11 and weight 2....
I. Manin symbols list.
(0 s)
II. 2-term relations.
(0.019 s)
III. 3-term relations.
Computing quotient by 4 relations.
(0.009 s)
(total time to create space = 0.029 s)
> SetVerbose("ModularSymbols",0);
Modular symbols can be input using Cusps().
> M := ModularSymbols(11,2);
> P := Cusps(); P;
Set of all cusps
> Type(P);
SetCsp
> oo := P!Infinity();
> M!<1,[oo,P!0]>; // note that 0 must be coerced into P.
{oo, 0}
> M!<1,[1/5,1]> + M!<1,[oo,P!0]>;
{-1/5, 0} + {oo, 0}
Modular symbols are also defined over finite fields.
> M := ModularSymbols(11,2,GF(7)); M; Full Modular symbols space of level 11, weight 2, and dimension 3 > BaseField(M); Finite field of size 7 > 7*M!<1,[1/5,1]>; 0