ABSTRACT. If ρ denotes a finite dimensional complex representation of SL 2 (Z), then it is known that the module M (ρ) of vector valued modular forms for ρ is free and of finite rank over the ring M of scalar modular forms of level one. This paper initiates a general study of the structure of M (ρ). Among our results are absolute upper and lower bounds, depending only on the dimension of ρ, on the weights of generators for M (ρ), as well as upper bounds on the multiplicities of weights of generators of M (ρ). We provide evidence, both computational and theoretical, that a stronger threeterm multiplicity bound might hold. An important step in establishing the multiplicity bounds is to show that there exists a free-basis for M (ρ) in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series E 6 of weight six.