  Let $q$ be a prime power and $\mathbb{F}_q$ be the finite field with $q$
elements. In this article we investigate the space of unramified automorphic
forms for $\mathrm{PGL}_n$ over the rational function field defined over
$\mathbb{F}_q$ (i.e.\ for $\mathbb{P}^1$ defined over $\mathbb{F}_q$). In
particular, we prove that the space of unramified cusp form is trivial and (for
$n=3$) that the space of eigenforms is one dimensional. Moreover, we show that
there are no nontrivial unramified toroidal forms for $\mathrm{PGL}_3$ over
$\mathbb{P}^1$ and conjecture that the space of all toroidal automorphic forms
is trivial.
