Finite-dimensional Banach spaces with numerical index zero

We prove that a finite-dimensional Banach space X has numerical index 0 if and only if it is the direct sum of a real space X-0 and nonzero complex spaces X-1,..., X-n in such a way that the equality parallel toX(0) + e(iq1p)X(1) + . . . + e(iqnp)Xnparallel to = parallel toX(0) + . . . + Xnparallel to holds for suitable positive integers q(1),. . . , q(n), and every p is an element of R and every x(j) is an element of X-j (j = 0, 1, . . . , n). If the dimension of X is two, then the above result gives X = C, whereas dim(X) = 3 implies that X is an absolute sum of R and C. We also give an example showing that, in general, the number of complex spaces cannot be reduced to one.