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Why the index of the principal congruence subgroup of level 2, defined as $$ \Gamma(2)=\left \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb{P}SL(2,\mathbb{Z}): \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv Id \mod 2 \right \} $$ has index $6$? And what are the representative cosets?

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There's a "reduction mod 2" map $$\operatorname{PSL}(2, \mathbf{Z}) \to \operatorname{PSL}(2, \mathbf{Z} / 2\mathbf{Z})$$ whose kernel is obviously $\Gamma(2)$. Moreover $\operatorname{PSL}(2, \mathbf{Z} / 2\mathbf{Z}) = \operatorname{SL}(2, \mathbf{Z} / 2\mathbf{Z}) = \operatorname{GL}(2, \mathbf{Z} / 2\mathbf{Z})$ and the latter has order 6 (easy to see by brute force, since there are only 16 possible 2x2 matrices over $\mathbf{Z} / 2\mathbf{Z}$, and if you write them down you'll see that 10 of them have determinant 0).

So the index of $\Gamma(2)$ is at most 6, and this upper bound is attained if and only if each of the elements of $\operatorname{SL}(2, \mathbf{Z} / 2\mathbf{Z})$ is the image of an element of $\operatorname{SL}(2, \mathbf{Z})$. Can you see how to check this?

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