  Let $\lambda(n)$ be the normalized n-th Fourier coefficient of a holomorphic
cusp form for the full modular group. We show that for some constant $C > 0$
depending on the cusp form and every fixed $c$ in the range $1 < c < 8/7$, the
mean value of $\lambda(p)$ is $\ll \exp (-C \sqrt{\log N})$ as p runs over all
(Piatetski-Shapiro) primes of the form $[n^c]$ with a natural number $n \leq
N$.
