Periodic PushASEP model is a bidirection interacting particle system with N particles moving on a torus of size L. To solve the system, we apply the Bethe Ansatz to compute the Fourier transform of the joint Markov process (X, Q) with respect to Q, where X is an N-tuple denoting the particle positions, and Q is the total current of the system. In particular, this can be written as a (N+1)-fold contour integral, which, by residue computations, simplifies into a 1-fold contour integral, then to a sum over Bethe roots. This leads to the two following applications: 1) The 1-fold contour integral is ready for asymptotic analysis, that we can perform at the relaxation time scale, giving similar limiting distribution as in [Baik and Liu, 2018]. 2) The sum over Bethe roots corresponds to the spectral decomposition of the system evolution, and when time t=0, it justifies rigorously the completeness of the Bethe eigenfunctions. Based on joint work with Axel Saenz (Oregon).