  In this article, the cohomology of the arithmetic group
$\mathrm{Sp}_4(\mathbb{Z})$ with coefficients in any finite dimensional highest
weight representation $\mathcal{M}_\lambda$ have been studied. Euler
characteristic with coefficients in $\mathcal{M}_\lambda$ have been carried out
in detail. Combining the results obtained on Euler characteristic and the work
of Harder on Eisenstein cohomology, the description of the cuspidal cohomology
has been achieved. At the end, we employ our study to compute the dimensions
for the cohomology spaces $H^{\bullet}(\mathrm{Sp}_4(\mathbb{Z}),
\mathcal{M}_\lambda)$.
