Hecke basis theorems for groups of genus 0

A Hecketype basis theorem is established for the cusp forms of negative even integral degree (m ultiplier syste m 1) on the class of Hecke groups. Hecke established the result for the classical modular group , which is the first of the Hecke groups. A second result is a parametrization theorem for en tire automorphic forms of negative real degree (with arbitrary multiplier systems) on certain discrete groups of real linear fractional transformations of genus zero.