[proofplan] We invoke the full Modularity Theorem for elliptic curves over $\mathbb Q$, proved by Wiles and Taylor-Wiles in the semistable case and by Breuil-Conrad-Diamond-Taylor in general. The argument is therefore a citation proof: we verify that the given object is exactly an elliptic curve over $\mathbb Q$, identify its conductor and good reduction primes, and then apply the external theorem to obtain the required weight $2$ normalized newform with matching Frobenius traces. [/proofplan] [step:Apply the external modularity theorem to the elliptic curve $E/\mathbb Q$] Let $N_E \in \mathbb N$ denote the conductor of the elliptic curve $E/\mathbb Q$. We use the external Modularity Theorem for elliptic curves over $\mathbb Q$, established in the semistable case by Wiles and Taylor-Wiles and completed in general by Breuil-Conrad-Diamond-Taylor; see Breuil-Conrad-Diamond-Taylor, [On the modularity of elliptic curves over $\mathbb Q$: wild 3-adic exercises](https://doi.org/10.1090/S0894-0347-01-00370-8). The theorem states that every elliptic curve over $\mathbb Q$ admits an associated weight $2$ normalized newform $f$ of level equal to its conductor $N_E$, such that the $L$-function of $E$ equals the $L$-function of $f$. The hypothesis of that theorem is exactly satisfied because the theorem statement gives that $E$ is an elliptic curve over $\mathbb Q$. Hence there exists a weight $2$ normalized newform $f$ of level $N_E$ associated to $E$. [/step] [step:Read off the Fourier coefficient at each good reduction prime] Let $p \in \mathbb N$ be a prime at which $E$ has good reduction. Define the Frobenius trace of $E$ at $p$ by \begin{align*} a_p(E) := p + 1 - |E(\mathbb F_p)|. \end{align*} For the normalized newform $f$ obtained from the Modularity Theorem, equality of the local Euler factors at every good reduction prime gives \begin{align*} a_p(f) = a_p(E). \end{align*} Substituting the definition of $a_p(E)$ yields \begin{align*} a_p(f) = p + 1 - |E(\mathbb F_p)|. \end{align*} Since $p$ was an arbitrary prime of good reduction, this proves the asserted coefficient identity for every such prime. [/step]