Serre's Conjecture Over Imaginary Quadratic Fields

The connection between classical modular forms and continuous representations of the absolute Galois group of the field of rational numbers has been the focus of major research in the last thirty years. In this thesis, we investigate a similar connection over the imaginary quadratic fields. We prove a weight reduction theorem for mod p modular forms and we prove the nonexistence of absolutely irreducible level 1 mod p Galois representations of small quadratic fields for p = 2, 3. We list certain conjectures, including an analogue of Serre's Conjecture. We present algorithms to compute the modular forms and the Hecke action on them. Irnplernenting the algorithms in MAGMA, we produce numerical evidence to support some of these conjectures.