Let $N \in \mathbb{N}$, and let $f \in S_2(\Gamma_0(N))$ be a normalized newform of exact level $N$ with Fourier expansion $f(q)=\sum_{n=1}^{\infty}a_n(f)q^n$ and $a_n(f)\in \mathbb{Q}$ for every $n\in\mathbb{N}$. Then there is an elliptic curve $E_f/\mathbb{Q}$ of conductor $N$ and a quotient morphism of abelian varieties over $\mathbb{Q}$

\begin{align*} \pi_f:J_0(N)\longrightarrow E_f \end{align*}

such that for every prime $p \nmid N$,

\begin{align*} a_p(E_f)&=a_p(f). \end{align*}

Moreover, $E_f$ is well-defined up to $\mathbb{Q}$-isogeny.