Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N \in \mathbb{N}$. Then there exists a normalized weight $2$ newform $f \in S_2(\Gamma_0(N))$ whose Hecke eigenvalues are rational and whose $L$-function equals the Hasse--Weil $L$-function of $E$:
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\begin{align*}
L(E,s)=L(f,s).
\end{align*}
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Equivalently, for every prime number $p$ with $p \nmid N$, if