Congruences and the Galois representations of classical cusp forms

Modular forms are central to modern number theory for many reasons, one of which being that they are a rich source of 2-dimensional Galois representations. But what information about the modular form is contained in the Galois representation? And how does one extract this information, for example about the Fourier coefficients of f?
In this expository talk, we will explore under what conditions the p-th Fourier coefficient of a classical normalised eigen cusp form f vanishing modulo some fixed prime ell is a congruence condition on the prime p, illustrating how to analyse and work with these kinds of Galois representations. This talk will focus on examples and will assume little background knowledge, so students and researchers from various mathematical disciplines are warmly invited to attend.