On the arithmetic of modular forms

In this short overview talk, we will stress the arithmetic significance of the coefficients of modular forms. This naturally leads to questions on the distribution of the coefficients in various senses. We will briefly touch on some of them and state some open questions.
The arithmetic information in the coefficients of a Hecke eigenform is summarised in the attached Galois representation. When studying its ramification properties, one notices that forms of weight one play a special role that we will explain.