A novel method for computing torus amplitudes for $\mathbb{Z}_{N}$ orbifolds without the unfolding technique

A novel method for computing torus amplitudes in orbifold compactifications is suggested. It applies universally for every Abelian $\mathbb{Z}_{N}$ orbifold without requiring the unfolding technique. This method follows from the possibility of obtaining integrals over fundamental domains of every Hecke congruence subgroup $\Gamma_{0}[N]$ by computing contour integrals over one-dimensional curves uniformly distributed in these domains.