For F-p the field on prime number p > 3 elements, it has been conjectured that there are just p + 3 evaluations of the Tutte polynomial in F-p which are computable in polynomial time. In this note it is shown that if p not equivalent to -1 mod 12 then there are further polynomial-time computable evaluations. (C) 2003 Elsevier Inc. All rights reserved.