Modular forms for triangle groups

Abstract: Modular forms are important in different areas of mathematics and theoretical physics. The theory is well known for the modular group PSL(2,Z), but is also of interest for other Fuchsian groups. In this thesis we will be interested in triangle groups with a cusp. We review some theory about mapping of hyperbolic triangles in order to derive an expression for the Hauptmodul of a triangle group, and use this to write a SageMath-program that calculates the Fourier series of the Hauptmodul. We then review some of the results presented in [4] that describe generalizations of well known concepts such as the Eisenstein series, the Serre derivative and some general results about the algebra of modular forms for triangle groups with a cusp. We correct some of the mistakes made in [4] and prove some further properties of the generators of the algebra of modular forms in the case of Hecke groups. Then we use the results from [4] to write a SageMath-program that calculates the Fourier series of the generators of the algebra of modular forms for triangle groups with a cusp and that also finds the relations between the generators in the special case of Hecke groups. Using the results from this program, we present some conjectures concerning the generators of the algebra of modular forms for a Hecke group, which, if proven to be true, give us a generalization of some of the Ramanujan equations. We conclude by explicitly calculating the generalized Ramanujan equations for the first few Hecke groups.