Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = \mathrm{PSL}_2(\mathbb R)$.

Up to a finite etale cover, $X$ is an etale cover of $Y(2)$. This is the same as saying that the fundamental group of $X$ is a subgroup of $P\Gamma(2)$ up to restricting to some finite index subgroup.

My question is whether it is really necessary to pass to an etale cover of $X$.

How do I construct an example of an $X$ as above for which $\pi_1(X)$ is not contained in $\Gamma(2)$?

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