Weil representations associated with finite quadratic modules

To any finite quadratic module, that is, a finite abelian group together with a non-degenerate quadratic form, it is possible to associate a representation of $mathrm{Mp}_{2}(mathbb Z )$ , the metaplectic cover of the modular group. This representation is usually referred to as a Weil representation and our main result is a general explicit formula for its matrix coefficients. This result completes earlier work by Scheithauer in the case when the representation factors through $mathrm{SL}_{2}(mathbb Z )$ . Furthermore, our formula is given in a such a way that it is easy to implement efficiently on a computer.