The first negative eigenvalue of Yoshida lifts

We prove that for any given $\epsilon >0$, the first negative eigenvalue of the Yoshida lift $F$ of a pair of elliptic cusp forms $f,g$ having square-free levels (where $g$ has weight $2$ and satisfies $(\log Q_{g})^2 \ll \log Q_f$), occurs before $c_{\epsilon} \cdot Q_F^{1/2-2 \theta+ \epsilon}$; where $Q_F,Q_f,Q_g$ are the analytic conductors of $F,f,g$ respectively, $\theta < 1/4$, and $c_{\epsilon}$ is a constant depending only on $\epsilon$.