  Let $\chi$ be a Dirichlet character modulo a prime~$p$. We give explicit
upper bounds on $q_1<q_2<\dots<q_n$, the $n$ smallest prime nonresidues of
$\chi$. More precisely, given $n_0$ and $p_0$ there exists an absolute constant
$C=C(n_0,p_0)>0$ such that $q_n\leq Cp^{\frac{1}{4}}(\log p)^{\frac{n+1}{2}}$
whenever $n\leq n_0$ and $p\geq p_0$.
