Recently Nicolas and Serre have determined the structure of the Hecke algebra acting on modular forms of level 1 modulo 2, and Serre has conjectured the existence of a universal Galois representation over this algebra. I'll explain the proof of this conjecture, and show how that representation may be used to get new information on the parity of the coefficients of modular forms of level 1 -- for example, on the parity of the values of the generalized Ramanujan's tau functions. I'll also explain a conjectural relation with the partition function.