  We study $h$-vectors and graded Betti numbers of level modules up to
multiplication by a rational number. Assuming a conjecture on the possible
graded Betti numbers of Cohen-Macaulay modules we get a description of the
possible $h$-vectors of level modules up to multiplication by a rational
number. We also determine, again up to multiplication by a rational number, the
cancellable $h$-vectors and the $h$-vectors of level modules with the weak
Lefschetz property. Furthermore, we prove that level modules of codimension
three satisfy the upper bound of the Multiplicity conjecture of Herzog, Huneke
and Srinivasan, and that the lower bound holds if the module, in addition, has
the weak Lefschetz property.
