On Hecke Eigenvalues at Piatetski-Shapiro Primes

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 $1<c<8\mathrm{/}7$, 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 $n \leq N$.