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We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_{k}(z)E_{\ell }(z)-E_{k+\ell }(z)$ in the standard fundamental domain lie on the boundary. We, moreover, find formulas for the number of zeros on the bottom arc with $|z|=1$, and those on the sides with $Re(z)=\pm 1/2$. One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.
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