  We study the moduli space of discrete, faithful, type-preserving
representations of the modular group $\mathbf{PSL}(2,\mathbb{Z})$ into
$\mathbf{PU}(3,1)$. The entire moduli space $\mathcal{M}$ is a union of
$\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$,
$\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ and some isolated
points. This is the first Fuchsian group such that its
$\mathbf{PU}(3,1)$-representations space has been entirely constructed. Both
$\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$ and
$\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ are parameterized
by a square, where two opposite sides of the square correspond to
representations of $\mathbf{PSL}(2,\mathbb{Z})$ into the smaller group
$\mathbf{PU}(2,1)$. In particular, both sub moduli spaces
$\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3} )$ and
$\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ interpolate the
geometries studied in \cite{FalbelKoseleff:2002} and \cite{Falbelparker:2003}.
