Let $\sigma$ be an element of $SL_{24}(\mathbb{Z})$ with its Jordan normal form is diagonal and the eigen values are $\epsilon_j$ for $1 \le j \le 24$ are n th root of unity where $n|N$ and $N$ is the finite order of $\sigma$. Equivalently we are describing $\sigma$ through its cycle shape $(a_1)^{b_1}\cdots(a_s)^{b_s}$.

We associate $\sigma$ to the following modular form:

$$\eta_{\sigma}(q) := \eta(\epsilon_1q)\cdots\eta (\epsilon_{24} q)=\eta(q^{a_1})^{b_1}\cdots \eta(q^{a_s})^{b_s}$$

Here $\eta$ stands for the Dedekind eta-function. Using the above defined $\eta_{\sigma}$ we define: $$\sum_{j>0}p_{\sigma}(1+j)q^{1+j} = \frac{q}{\eta_{\sigma}(q)}$$ This is a generalized partition function.

We assume the cycle type of $\sigma$ is $1^{1}23^{1}$ and hence $N = 23$. In this case I have the following questions :

What is this generalized partition function $p_{\sigma}$ and how to find $p_{\sigma}(n)$ for some natural number $n$?

There are many generalisations of partitions functions and hence what is the reference for this particular type of generalized partition function?

Thanks for your time.

Have a good day.