[proofplan] We use the conductor-preserving form of the modularity theorem for elliptic curves over $\mathbb{Q}$, proved by Breuil, Conrad, Diamond, and Taylor after the work of Wiles and Taylor-Wiles. That theorem asserts that the $L$-function of any elliptic curve over $\mathbb{Q}$ is the $L$-function of a weight $2$ normalized newform whose level is the conductor of the curve. Comparing the Euler factors at primes of good reduction then identifies the newform coefficient $a_p(f)$ with the Frobenius trace $p+1-|E(\mathbb{F}_p)|$. [/proofplan] [step:Name the conductor and the good-prime Frobenius traces] Let $N_E \in \mathbb{N}$ denote the conductor of the elliptic curve $E/\mathbb{Q}$. For each prime $p$ at which $E$ has good reduction, define the Frobenius trace $a_p(E) \in \mathbb{Z}$ by \begin{align*} a_p(E) := p + 1 - |E(\mathbb{F}_p)|. \end{align*} For such a prime $p$, let $L_p(E,s)$ denote the local Euler factor at $p$ of the Hasse-Weil $L$-function of $E$. Then $a_p(E)$ is the coefficient appearing in that good Euler factor: \begin{align*} L_p(E,s)^{-1} = 1 - a_p(E)p^{-s} + p^{1-2s}. \end{align*} [/step] [step:Apply the conductor-preserving modularity theorem] By the conductor-preserving modularity theorem for elliptic curves over $\mathbb{Q}$, proved in [Modularity of elliptic curves over $\mathbb{Q}$ by Breuil, Conrad, Diamond, and Taylor](https://doi.org/10.1090/S0894-0347-01-00370-8), every elliptic curve $E/\mathbb{Q}$ is associated to a weight $2$ normalized newform of level $N_E$ whose $L$-function equals the Hasse-Weil $L$-function of $E$. Applying this theorem to the given elliptic curve $E/\mathbb{Q}$ gives a weight $2$ normalized newform \begin{align*} f: \mathbb{H} &\to \mathbb{C} \end{align*} of level $N_E$, where $\mathbb{H} := \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}$ is the complex upper half-plane, such that $L(f,s) = L(E,s)$ including the local Euler factors. For each prime $p$, let $L_p(f,s)$ denote the local Euler factor at $p$ of the newform $L$-function $L(f,s)$. [/step] [step:Compare good Euler factors to identify the Fourier coefficients] Write the Fourier expansion of the normalized newform $f$ at infinity as \begin{align*} f(z) = \sum_{n=1}^{\infty} a_n(f)e^{2\pi i n z}, \qquad z \in \mathbb{H}, \end{align*} with $a_1(f)=1$. For elliptic curves over $\mathbb{Q}$, the primes of bad reduction are exactly the primes dividing the conductor $N_E$; equivalently, a prime $p$ is of good reduction if and only if $p \nmid N_E$. Hence, for a prime $p$ of good reduction, we have $p \nmid N_E$, and the local Euler factor of the newform is \begin{align*} L_p(f,s)^{-1} = 1 - a_p(f)p^{-s} + p^{1-2s}. \end{align*} Since $L(f,s)=L(E,s)$ with equality of local Euler factors, the two quadratic polynomials in $p^{-s}$ agree: \begin{align*} 1 - a_p(f)p^{-s} + p^{1-2s} = 1 - a_p(E)p^{-s} + p^{1-2s}. \end{align*} Comparing the coefficients of $p^{-s}$ gives $a_p(f)=a_p(E)$. By the definition of $a_p(E)$, this is \begin{align*} a_p(f)=p+1-|E(\mathbb{F}_p)|. \end{align*} Thus the required weight $2$ normalized newform of level equal to the conductor of $E$ exists and has the stated good-prime coefficients. [/step]