Bounds for the first several prime character nonresidues

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^{\frac 14+\epsilon } with \chi (\ell )=1.