The sum of the modular forms f and g.
The sum of the modular form f and the power series g.
The q-expansion of f must be coercible into the
parent of g. The sum g + f is also defined, as are
the differences f - g and g - f.
The difference of the modular forms f and g.
The product of the scalar a and the modular form f.
The product of the scalar 1/a and the modular form f.
The power fn of the modular form f, where n≥1 is
an integer.
The product of the modular forms f and g.
The only condition is that the base fields of f and g
be the same. The weight of f*g is the sum
of the weights of f and g.
> M2 := ModularForms(Gamma0(11), 2);
> f := M2.1;
> g := M2.2;
> f;
1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + 24*q^6 + 24*q^7 + O(q^8)
> g;
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
> f+g;
1 + q + 10*q^2 + 11*q^3 + 14*q^4 + 13*q^5 + 26*q^6 + 22*q^7 + O(q^8)
> 2*f;
2 + 24*q^2 + 24*q^3 + 24*q^4 + 24*q^5 + 48*q^6 + 48*q^7 + O(q^8)
> MQ,phi := BaseExtend(M2, RationalField());
> phi(2*f)/2;
1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + 24*q^6 + 24*q^7 + O(q^8)
> f^2;
1 + 24*q^2 + 24*q^3 + 168*q^4 + 312*q^5 + 480*q^6 + 624*q^7 + O(q^8)
> Parent($1);
Space of modular forms on Gamma_0(11) of weight 4 and dimension 4 over
Integer Ring.
> M3 := ModularForms([DirichletGroup(11).1], 3); M3;
Space of modular forms on Gamma_1(11) with character all conjugates of
[$.1], weight 3, and dimension 3 over Integer Ring.
> M3.1*f;
1 + 12*q^2 + 2*q^3 + 6*q^4 - 126*q^5 - 168*q^6 - 384*q^7 + O(q^8)
> Parent($1);
Space of modular forms on Gamma_1(11) of weight 5 and dimension 25
over Integer Ring.
V2.28, 13 July 2023