  For a simple-root $\lambda$-constacyclic code $\mathcal{C}$ over
$\mathbb{F}_q$, let $\langle\rho\rangle$ and $\langle\rho,M\rangle$ be the
subgroups of the automorphism group of $\mathcal{C}$ generated by the cyclic
shift $\rho$, and by the cyclic shift $\rho$ and the scalar multiplication $M$,
respectively. Let $N_G(\mathcal{C}^\ast)$ be the number of orbits of a subgroup
$G$ of automorphism group of $\mathcal{C}$ acting on
$\mathcal{C}^\ast=\mathcal{C}\backslash\{0\}$. In this paper, we establish
explicit formulas for $N_{\langle\rho\rangle}(\mathcal{C}^\ast)$ and
$N_{\langle\rho,M\rangle}(\mathcal{C}^\ast)$. Consequently, we derive a upper
bound on the number of nonzero weights of $\mathcal{C}$. We present some
irreducible and reducible $\lambda$-constacyclic codes, which show that the
upper bound is tight. A sufficient condition to guarantee
$N_{\langle\rho\rangle}(\mathcal{C}^\ast)=N_{\langle\rho,M\rangle}(\mathcal{C}^\ast)$
is presented.
