  Let $\varepsilon > 0$. We prove that there are constants
$m_0=m_0(\varepsilon)$ and $\kappa=\kappa(\varepsilon) > 0$ for which the
following holds: For every integer $m > m_0$ and every nontrivial Dirichlet
character modulo $m$, there are more than $m^{\kappa}$ primes $\ell \le
m^{\frac{1}{4\sqrt{e}}+\varepsilon}$ with $\chi(\ell)\notin \{0,1\}$. The proof
uses the fundamental lemma of the sieve, Norton's refinement of the Burgess
bounds, and a result of Tenenbaum on the distribution of smooth numbers
satisfying a coprimality condition. For quadratic characters, we demonstrate a
somewhat weaker lower bound on the number of primes $\ell \le
m^{\frac14+\epsilon}$ with $\chi(\ell)=1$.
