  We produce nontrivial asymptotic estimates for shifted sums of the form $\sum
a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier
coefficients of holomorphic cusp forms. These results are unconditional, but we
demonstrate how to strengthen them under the Riemann Hypothesis. As an
application, we show that there are infinitely many three term arithmetic
progressions $n-h, n, n+h$ such that $a(n-h)a(n)a(n+h) \neq 0$.
