On the cuspidal modular forms for the Fuchsian groups of the first kind

In this paper we study the construction and non-vanishing of cuspidal modular forms of weight m⩾3 for arbitrary Fuchsian groups of the first kind. We give a spanning set for the space of cuspidal modular forms Sm(Γ) of weight m⩾3 in a uniform way which does not depend on the fact that Γ has cusps or not.