Upper bound for $\mathbb{E}[e^{-aX}]$, where X follows binomial distribution $B(n,p)$

Question

Here $a$ is a postive constant. $p$ is also fixed. Since $e^{-aX}$ is convex, it's easy to get $\mathbb{E}[e^{-aX}] \ge e^{-a\mathbb{E}[X]} = e^{-anp}$. However, what I need is one upper bound. Is $\mathbb{E}[e^{-aX}]$ bounded by one exponential function of $n$ ( like $e^{-bn}$)?