Bound on coefficients of Hasse-Weil function

Question

Let $E$ be an elliptic curve over $\mathbb{Q}$. For $n\in \mathbb{N}$, we define $a_n$ as follows.

1. $a_1 = 1$.
2. If $n = p$, $$ a_p = \begin{cases} p+1-|E(\mathbb{F_p})| & E \text{ has good reduction at } p\\ 0 & E \text{ has additive reduction at } p\\ 1 & E \text{ has split multiplicative reduction at } p\\ -1 & E \text{ has non split multiplicative reduction at } p\\ \end{cases}. $$
3. If $n=p^r$, for $r\ge 2$ $$ a_{p^r} = \begin{cases} a_p\cdot a_{p^{r-1}} - p\cdot a_{p^{r-2}} & E \text{ has good reduction at } p\\ (a_p)^r & E \text{ has bad reduction at } p\\ \end{cases}. $$
4. If $n=n_1n_2$, with $(n_1,n_2) = 1$, then $a_n = a_{n_1}\cdot a_{n_2}$.

Then the Hasse-Weil function can be written as the series $$ L(E,s) = \sum_{n\in\mathbb{N}}\frac{a_n}{n^s}. $$ My question relates to the Hasse bound which says $|a_p|\le 2\sqrt{p}$. Can a similar bound be extended to all $a_n$? My main obstacle is the recursive definition of $a_{p^r}$.