Bounds for the first several prime character nonresidues

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Let ε > 0. We prove that there are constants m₀=m₀(ε) and κ=κ(ε) > 0 for which the following holds: For every integer m > m₀ and every nontrivial Dirichlet character modulo m, there are more than mκ primes ℓ ≤ m14√e+ε with χ(ℓ)∉ {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 ℓ ≤ m14+ε with χ(ℓ)=1.