  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 $\ell < 100$ and all integers $m \geq 1$, we have $$ \tau(n)
\neq \pm \ell, \pm 5^m. $$ 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 $a_f(p^2)$ is never a $j$-th power. We prove a
similar result for $a_f(p^4)$, 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.
