In the Zagier's chapter of 1-2-3 of Modular Forms he stated as obvious (or just known?) fact that for $F(z) = (z^{k-i}f(z))_{i=0, ..., k}$ and $g = \begin{pmatrix}a & b \\ c & d \end{pmatrix} \in \operatorname {SL}_2(\mathbb Z)$ there is an identity
$(az + b)^{k-m}(cz+d)^mf(z) = \sum^m_{n=0}M_{mn}F_n(z)$
where $M \in \operatorname {SL}(k+1, Z)$ is the $k-$th symmetric power of $g$.
I've never encountered the symmetric powers before and I'm really confused what's going on here, as I cannot really make the connection between that identity from above and definitions or properties of symmetric powers that I've found