Let k be an algebraic closed field of characteristic zero. Macaulay's inverse system gives a bijection between finitely generated R-submodules of k[x_1 , . . . , x_n ], and ideals I in R such that R/I is Artinian. This correspondence can be restricted to suitable finitely generated R- modules and ideals I such that R/I is level, i.e. such that all socle elements in R/I have the same order. We use this tool to characterize h-vectors which are admissible for level local algebras (R/I, m, k) such that m^4 = 0, and to prove that level local algebras with maximal h-vector and m^4 = 0 are in fact graded.