  We consider the problem of determining the cross-correlation values of the
sequences in the families comprised of constant multiples of $M$-ary Sidelnikov
sequences over $\mathbb{F}_q$, where $q$ is a power of an odd prime $p$. We
show that the cross-correlation values of pairs of sequences from such a family
can be expressed in terms of certain Jacobi sums. This insight facilitates the
computation of the cross-correlation values of these sequence pairs so long as
$\phi(M)^{\phi(M)} \leq q.$ We are also able to use our Jacobi sum expression
to deduce explicit formulae for the cross-correlation distribution of a family
of this type in the special case that there exists an integer $x$ such that
$p^x \equiv -1 \pmod{M}.$
