Inspired by an answer to

What is the intuition between 1-cocycles (group cohomology)?,

one may wonder what are the meromorphic functions $f \in \mathcal{M}(\mathbb{C})$ for which there exists a function $j = j(f): SL_2(\mathbb{Z}) \times \mathbb{C} \rightarrow \mathbb{C}$ that satisfies

$$f\left(\frac{az + b}{cz + d}\right) = j(a, b, c, d, z)f(z),$$ when $ad - bc = 1$. In particular, are all such $f$ necessarily modular? Also, what is the cohomology group

$$H^1(SL_2(\mathbb{Z}), \mathcal{M}(\mathbb{C}))?$$

(I suspect this might be known or addressed somewhere).



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