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how do I compute the fundmental domain for a congruence subgroup of $SL(2, \mathbb{Z})$

This region is important because the theta function $\theta(z) = q^{n^2}$ is invariant under two transformations $z \mapsto z + 1$ and $z \to - \frac{1}{4z}$, and these two generate the congruence subgroup $\Gamma_0(4)$.

I know there is an algorithm for any Fuchsian group. Here is an erroneous comutation of the fundamental domain. It has too many cusps.

I found a blog with a picture of what could be the fundamental domain of $\Gamma_0(2)$. Therefore the domain for $\Gamma_0(4)$ could be related. I know that $[SL(2,\mathbb{Z}): \Gamma_0(2)]= 3$ and $[SL(2,\mathbb{Z}): \Gamma_0(4)]= 6$ but I can't figure out which two copies to join. And ther's no derivation for this particular case.

enter image description here

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    It would help if your question contained a bit more information such as the definition of $\Gamma_0(4)$. – 2017-02-26
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    Use the fundamental domain for SL(2,$\mathbb Z$), and look at the cosets for the congruence subgroup. (I think I illustrated this procedure in my modular forms notes, for example.) – 2017-02-26
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    @kimball can you provide link? I think there's an easy trick for congruence groups, I am reading out of notes of Zagier who leaves it as exercise. I need area of fundamental region in order to show there's only one modular form of weight 4. – 2017-02-26
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    My notes are [here](http://www.math.ou.edu/~kmartin/mfs/), see Example 3.4.11 for the case of $\Gamma_0(2)$. I leave $\Gamma_0(4)$ as an exercise. But it has index 6 in SL(2,$\mathbb Z$) so the area will be 6 times that of a fundamental domain for SL(2,$\mathbb Z$). See also http://math.stackexchange.com/q/1316464/11323 – 2017-02-26

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