In this paper, a general method to obtain constructive proofs of existence of periodicorbits in the forced autonomous Navier–Stokes equations on the three-torus isproposed. After introducing a zero finding problem posed on a Banach space of geometricallydecaying Fourier coefficients, a Newton–Kantorovich theorem is applied toobtain the (computer-assisted) proofs of existence. The required analytic estimates toverify the contractibility of the operator are presented in full generality and symmetriesfrom themodel are used to reduce the size of the problem to be solved. As applications,we present proofs of existence of spontaneous periodic orbits in the Navier–Stokesequations with Taylor–Green forcing.