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1 [made Community Wiki]

If $n \ge 3$, perhaps one can use Margulis superrigidity to deduce that any homomorphism from $\mathrm{SL}(n,\mathbb{Z})$ to $\mathrm{GL}(m,\mathbb{C})$ extends to a representation of $\mathrm{SL}(n,\mathbb{C})$. The standard representation of $\mathrm{SL}(n,\mathbb{Z})$ factors through $\mathrm{PSL}(n,\mathbb{Z})$ if and only if $n$ is odd. The next smallest representation of $\mathrm{SL}(n,\mathbb{C})$ has dimension $> n+1$, (assuming that $n > 2$). If $n = 2$, then $\mathrm{PSL}(2,\mathbb{Z})$ is certainly a subgroup of $\mathrm{GL}(3,\mathbb{Z})$. So the answer seems to be: if and only if $n = 2$ or $n$ is odd.