  Suppose that $M$ is an even lattice with dual $M^{*}$ and level $N$. Then the
group $Mp_{2}(\mathbb{Z})$, which is the unique non-trivial double cover of
$SL_{2}(\mathbb{Z})$, admits a representation $\rho_{M}$, called the Weil
representation, on the space $\mathbb{C}[M^{*}/M]$. The main aim of this paper
is to show how the formulae for the $\rho_{M}$-action of a general element of
$Mp_{2}(\mathbb{Z})$ can be obtained by a direct evaluation which does not
depend on ``external objects'' such as theta functions. We decompose the Weil
representation $\rho_{M}$ into $p$-parts, in which each $p$-part can be seen as
subspace of the Schwartz functions on the $p$-adic vector space
$M_{\mathbb{Q}_{p}}$. Then we consider the Weil representation of
$Mp_{2}(\mathbb{Q}_{p})$ on the space of Schwartz functions on
$M_{\mathbb{Q}_{p}}$, and see that restricting to $Mp_{2}(\mathbb{Z})$ just
gives the $p$-part of $\rho_{M}$ again. The operators attained by the Weil
representation are not always those appearing in the formulae from 1964, but
are rather their multiples by certain roots of unity. For this, one has to find
which pair of elements, lying over a matrix in $SL_{2}(\mathbb{Q}_{p})$, belong
to the metaplectic double cover. Some other properties are also investigated.
