This paper is the numerical implementation of the very elegant work done by Forrest, MacLaughlin and Ercloni in which they develop a perturbation theory for periodic sine-Gordon. The nonlinear normal modes for the solitons in this case are represented by the motion of pairs of points on a multiply-connected Riemann surface. Solution of the equations requires the integration along curves on this Riemann surface. The merging of points corresponds to motion through a homoclinic orbit. The results obtained from the reduced set of ODEs agrees very well with full simulations of the PDEs from which the spectra are calculated via a forward numerical spectral transformation.