Zeroth Bessel function has the following integral representation: $$ J_0 (x) = \int_0^{\pi} \frac{dz}{\pi} e^{ix\cos z} $$
Does the nth power of the zeroth Bessel function $J_0 (x)^n$ also have similar integral representation ($n \in N$)? I would like to find the integral representation of $J_0 (x)^n$ in the simplest form possible (It would be nice if it involved only a single integral), but it is also fine if the obtained expressions are a little more complicated.