Higher order Maass forms

We determine the size of spaces of higher order Maass forms of even weight for cofinite discrete subgroups of PSL(2,R) with cusps. If exponential growth at the cusps is allowed, the spaces of Maass forms of a given order are as large as algebraic constrictions allow. We show the analogous statement for the spaces of holomorphic forms. For functions on the universal covering group of PSL(2,R) we introduce the concept of generalized weight. For the resulting spaces of higher order Maass forms with even generalized weight we show that the size is maximal.