  We describe a new method for constraining Laplacian spectra of hyperbolic
surfaces and 2-orbifolds. The main ingredient is consistency of the spectral
decomposition of integrals of products of four automorphic forms. Using a
combination of representation theory of $\mathrm{PSL}_2(\mathbb{R})$ and
semi-definite programming, the method yields rigorous upper bounds on the
Laplacian spectral gap. In several examples, the bound is nearly sharp. For
instance, our bound on all genus-2 surfaces is $\lambda_1\leq 3.8388976481$,
while the Bolza surface has $\lambda_1\approx 3.838887258$. The bounds also
allow us to determine the set of spectral gaps attained by all hyperbolic
2-orbifolds. Our methods can be generalized to higher-dimensional hyperbolic
manifolds and to yield stronger bounds in the two-dimensional case. The ideas
were closely inspired by modern conformal bootstrap.
