Arithmetic#
- f + g: ModFrmElt, ModFrmElt -> ModFrmElt#
The sum of the modular forms \(f\) and \(g\).
- f + g: ModFrmElt, RngSerPowElt -> RngSerPowElt#
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 \(g-f\).
- f - g: ModFrmElt, ModFrmElt -> ModFrmElt#
The difference of the modular forms \(f\) and \(g\).
- a * f: RngElt, ModFrmElt -> ModFrmElt#
The product of the scalar \(a\) and the modular form \(f\).
- f / a: ModFrmElt, RngElt -> ModFrmElt#
The product of the scalar \(1/a\) and the modular form \(f\).
- f ^ n: ModFrmElt, RngIntElt -> ModFrmElt#
The power \(f^n\) of the modular form \(f\), where \(n\geq 1\) is an integer.
- f * g: ModFrmElt, ModFrmElt -> ModFrmElt#
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*gis the sum of the weights of \(f\) and \(g\).
- Example: Arithmetic (ex-061f7d)#
> 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.