An estimate of Gaussian sums

Suppose $n\in N$, $n\ge3$, $a\in Z$, $q\in N$, $(a,q)=1$. It is shown that for the Gaussian sums
$$ S_n(a,q)=\sum_{k=0}^{q-1}e^{2\pi i\frac aqk^n} $$
the following estimate holds uniformly with respect to all parameters:
$$ |S_n(a,q)|\le\exp\{C(n\varphi(n))^2\}q^{1-1/n}, $$
where $C$ is a positive absolute constant and $\varphi(n)$ is Euler's function.