We all know the full modular (cusp) form of weight 12 $$ \Delta(z) = \sum_{n=1} \tau(n)q^n = q \prod_{n=1} (1-q^n)^{24} $$ that generates the multiplicative Ramunujan tau function $\tau(n)$.
Today I was thinking about following: Take three constants $\alpha,\beta,\gamma \in \mathbb{N}$ and generate $f_{\alpha,\beta,\gamma }(n)$ with $$ \sum_{n=1} f_{\alpha,\beta,\gamma }(n)q^n = q \prod_{n=1} (1-\alpha q^{\beta n})^\gamma .$$
Clearly $f_{1,1,24}(n) = \tau(n)$.
Now I asked the questions, are there other $f$ that are multiplicative? Via bruteforce I found an interesting result, that is:
Let $x \cdot y = 24$. Then $f_{1,x,y}(n)$ is multiplicative, i.e. $$ \sum_{n=1} f_{1,x,y }(n)q^n = q \prod_{n=1} (1-q^{x n})^y .$$ Examples: $~q \prod_{n=1} (1-q^{2 n})^{12}~$, $~q \prod_{n=1} (1-q^{3 n})^8~$ or $~q \prod_{n=1} (1-q^{24 n})$.
This seems quite interesting to me. My question now: Can someone give a simple explanation or refer to some paper where this was mentioned?
Furthermore: I assume that besides $f_{1,1,24}$ no other $f_{\alpha,\beta,\gamma}$ is a modular form. Is that correct?