  We study generic one-loop (string) amplitudes where an integration over the
fundamental region F of the modular group is needed. We show how the known
lattice-reduction technique used to unfold F to a more suitable region S can be
modified to rearrange generic modular invariant amplitudes. The main aim is to
unfold F to the strip and, at the same time, to simplify the form of the
integrand when it is a sum over a finite number of terms, like in one-loop
amplitudes for closed strings compactified on orbifolds. We give a general
formula and a recipe to compute modular invariant amplitudes. As an application
of the technique we compute the one-loop vacuum energy \rho_n for a generic
\Z_n freely acting orbifold, generalizing the result that this energy is less
than zero and drives the system to a tachyonic divergence, and that
\rho_n<\rho_m if n>m.
