Self-reciprocal irreducible polynomials with prescribed coefficients

We prove estimates for the number of self-reciprocal monic irreducible polynomials over a finite field of odd characteristic, that have the t lower degree coefficients fixed to given values. Our estimates imply that one may specify up to m/2−logq(2m)−1 values in the field and a self-reciprocal monic irreducible polynomial of degree 2m exists with its low degree coefficients fixed to those values.