Arithmetic, analytic, and geometric aspects of the theory of modular forms

This thesis will address several different aspects of the theory of modular forms. The first topic we will consider is the classical Eichler-Selberg trace formula. This is an explicit formula for the trace of the Hecke operator Tn acting on the vector space Sk(Gamma 0(N)). We consider the problem of when (as a function of n, k and N) this trace is equal to zero.; The second topic we will consider is the distribution of the Fourier coefficients of newforms f without complex multiplication. This distribution is closely related to the predicted analytic properties of the symmetric power L-functions associated to f. We will show that the Fourier coefficients of f can rarely be "too small," conditional on standard conjectures about the symmetric power L-functions.; The third topic we will consider are some generalizations of work of Zagier. Zagier proved a striking identity relating the Fourier coefficients of modular forms in two different spaces. Zagier used this identity to establish a relationship between these coefficients and traces of singular moduli. We will consider generalizations of these two results to the context of Hilbert modular forms.