We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes ℓ<100 and all integers m≥1, we haveτ(n)≠±ℓ,±5m. For general newforms f with even integer weight 2k and integer coefficients, we prove for most integers j dividing 2k−1 and all ordinary primes p that af(p2) is never a j-th power. We prove a similar result for af(p4), conditional on the Frey-Mazur Conjecture. Our primary method involves relating questions about values of newforms to the existence of perfect powers in certain binary recurrence sequences, and makes use of bounds from the theory of linear forms in logarithms. The method extends without difficulty to a large family of Lebesgue-Nagell equations with fixed exponent. To prove results about general newforms, we also make use of the modular method and Ribet's level-lowering theorem.