  We prove a higher order asymptotic formula for traces of finite block
Toeplitz matrices with symbols belonging to H\"older-Zygmund spaces. The
remainder in this formula goes to zero very rapidly for very smooth symbols.
This formula refine previous asymptotic trace formulas by Szeg\"o and Widom and
complement higher order asymptotic formulas for determinants of finite block
Toeplitz matrices due to B\"ottcher and Silbermann.
