Ruling polynomials and augmentations over finite fields

For any Legendrian link, L, in (\R^3, \ker(dz-y\,dx)) we define invariants, Aug_m(L,q), as normalized counts of augmentations from the Legendrian contact homology DGA of L into a finite field of order q where the parameter m is a divisor of twice the rotation number of L. Generalizing a result of Ng and Sabloff for the case q =2, we show the augmentation numbers, Aug_m(L,q), are determined by specializing the m-graded ruling polynomial, R^m_L(z), at z = q^{1/2}-q^{-1/2}. As a corollary, we deduce that the ruling polynomials are determined by the Legendrian contact homology DGA.