Virasoro conformal bootstrap with $c > 1$

We derive new constraints on the spectrum of two-dimensional conformal field theories with central charge $c>1.$ Employing the pillow representation of the four point correlator of identical scalars with dimension $\Delta_{\mathcal{O}}$ and positivity of the coefficients of its expansion in the elliptic nome we place central charge dependent bounds on the dimension of the first excited Virasoro primary the scalar couples to, in the form $\Delta_1<f(c,\Delta_{\mathcal{O}}).$ We give an analytic expression for $f(c,\Delta_{\mathcal{O}})$ and write down transcendental equations that significantly improve the analytic bound. We numerically evaluate the stronger bounds for arbitrary fixed values of $c$ and $\Delta_{\mathcal{O}}.$