We develop a new method to bound the hyperbolic and spherical Fourier coefficients of Maass forms defined with respect to arbitrary uniform lattices.

Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup-norm of a half-integral weight cusp form is bounded in terms of the level: we prove that $\left|\right|y^{\kappa /2}{f̃\left|\right|}_{\infty }{\ll }_{\epsilon ,\kappa }{N}^{1/2-1/18+\epsilon }\left|\right|y^{\kappa /2}f̃{\left|\right|}_{L^2}$ for a modular form f̃ of level 4N and weight κ, a half-integer.