Interesting multiplicative Ramanujan-like q-expansions

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?

• Your $f_{1,3,8}$ is the generating function of OEIS sequence A000731 and it is a cusp form of $\Gamma_0(9)$. – Somos Sep 12 '17 at 14:25
• You can look in OEIS for more examples. Note Yves Martin, "Multiplicative $\eta$-quotients", Trans. A.M.S. 1996 have some but not all of them. – Somos Sep 12 '17 at 14:32
• In general to prove it is multiplicative, I would show it is a cusp form in $S_k(\Gamma)$, find $d$ the dimension of $S_k(\Gamma)$, construct a basis (theta functions, Eisenstein series, double coset operator ?) find the $d$ linear combinations whose first few coefficients are multiplicative. And magma knows how to automatize this process. @Somos – reuns Sep 12 '17 at 23:45