be a normalized newform of level $N$ whose Fourier coefficients lie in $\mathbb{Q}$. For each prime $p\nmid N$, let
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\begin{align*}
a_p(E) := p + 1 - \#E(\mathbb{F}_p)
\end{align*}
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be the Frobenius trace of $E$ at $p$. Suppose that
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\begin{align*}
a_p(E) = a_p(f)
\end{align*}
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for every prime $p\nmid N$, and suppose that, for every prime $p\mid N$, the local Euler factor of the Hasse-Weil $L$-function $L(E,s)$ equals the local Euler factor of the newform $L$-function $L(f,s)$. Then
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\begin{align*}
L(E,s)=L(f,s)
\end{align*}
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as meromorphic functions of $s\in\mathbb{C}$. Consequently, $E$ is modular, with associated newform $f$.