Example ModSym_Creation (H145E1)

We compute a basis for the space of modular symbols of weight 2, level 11 and trivial character.

> M := ModularSymbols(11,2); M;
Full modular symbols space for Gamma_0(11) of weight 2 and dimension 3
over Rational Field
> 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 for Gamma_0(11) of weight 2 and dimension 3
over Finite field of size 7
> BaseField(M);
Finite field of size 7
> 7*M!<1,[1/5,1]>;
0

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