Sifted character sums and free quotients of Bianchi groups

We show that the Bianchi group $\mathrm{PSL}_2(\mathcal{O}_d)$, where $\mathcal{O}_d$ is the ring of integers in $\Q(\sqrt{d})$, $d<0$, has a free quotient of rank $\geq |d|^{1/4-\epsilon}$, as $|d|\to\infty$.