Let $\Gamma(2)$ be the subgroup of $\mathrm{SL}_2(\mathbf{Z})$ satisfying the usual congruence conditions. It acts on the complex upper half-plane.
Does it have a fundamental domain contained in the strip $$\{x+iy: -1\leq x \leq 1\}?$$
Is the following a correct argument?
The matrix $$\left( \begin{matrix} 1 & \pm 2 \\ 0 & 1 \end{matrix}\right)$$ is in $\Gamma(2)$. If $\tau$ is not in the above strip, we can translate $\tau$ into this strip by multiplying with the above matrix a couple of times.