I was hoping that someone could help clarify a source of confusion for me, I must be doing and saying something wrong but I just don't know what: Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $$L(s,E)=\sum_{n=1}^{\infty}a_n(E)n^{-s}$$ be the Hasse-Weil $L$-function of $E$. Finally, let $\tilde{E}$ be the reduction of $E$ mod $p$ and assume that $p$ is a prime for which $E$ has good reduction.

Then $$a_p(E)=p+1-|\tilde{E}(\mathbb{F}(p))|$$ and setting $a_1(E)=1$ the $p$ power coefficients are given by $$a_{p^e}(E)=a_p(E)a_{p^{e-1}}(E)-pa_{p^{e-2}}(E).$$

Now looking at Diamond and Shurman, for instance, I find that also we can write $$a_{p^e}(E)=p^e+1-|\tilde{E}(\mathbb{F}(p^e))|$$ but when I use this expression as a "definition" of $a_{p^e}(E)$ and do some explicit calculations I don't get the right recursion, for instance I seem to get in practice $$a_{p^2}(E)=a_p(E)^2 - 2p$$ instead of $$a_{p^2}(E)=a_p(E)^2-p.$$

I must be misunderstanding something, but I can't figure out what. Any help?