TITLE: Alternate expresion of L-series coefficients
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: The mistake is acknowledged and corrected in the errata for the third printing:
http://people.reed.edu/~jerry/MF/mferrata3.pdf