I know that my question is very simple, but I can not handle it myself. Also, I do not know whether it is a suitable question for this site or not, because I want to calculate with Magma. I read the other questions about "eta_products" (I can't send linky texts because "my karma is insufficient to publish links"), but at the moment I do not have a system of my own on which I am allowed to install and run Sage.

I want to calculate the $n^{\text{th}}$-Fourier coefficient of some functions that are closed to $\eta$-quotients, in fact some functions that are very similar to Euler function, but necessarily the same.

For instance, given positive integers $k, K, l, L$, with $k \mid K$ and $l \mid L$. I want to calculate the $n^{\text{th}}$-Fourier coefficient of $\dfrac{\phi(q^K)\phi(q^L)}{\phi(q^k)\phi(q^l)}$ or $\dfrac{\phi(q)\phi(q^K)\phi(q)\phi(q^L)}{\phi(q^k)\phi(q^l)}$ or some other types of $\dfrac{\prod_{i=1}^{T}\phi(q^{L_i})^{A_i}}{\prod_{i=1}^{t}\phi(q^{l_i})^{a_i}}$ for suitable positive integers, and some other quotients that are based on some other functions that have some similarity with Euler function.

**My question**: Now forget Euler function, and whatever I've said.

If I know how can I calculate the $n^{\text{th}}$-Fourier coefficient of $$\dfrac{\prod_{m=1}^{\infty}\Bigg((1-q^K)(1-q^L)\Bigg)}{\prod_{m=1}^{\infty}\Bigg((1-q^k)(1-q^l)\Bigg)},$$ then I can calculate the $n^{\text{th}}$-Fourier coefficient of the others.

**My attempts in Magma** (I've tried, but I don't know how to write these codes in beautiful boxes here):

- I can calculate the coefficients of a polynomial:
`Coefficient(5*x^4+4*x^3+3*x^2+2*x+1, 2);`

- also, I can do some other things like:
`PQ<x> := PowerSeriesRing(RationalField()); f := (1-x) / (1-x^2); Coefficients(f);`

- also, I can define this (but this isn't going to work):
`f := func< N, q | &*[1 - q^(i) : i in [1..N]] >;`

- This code almost works:
`PQ<x> := PowerSeriesRing(RationalField()); N :=100; f := func< x | &*[(1 - x^(i)) : i in [1..N]] >; F :=(f(x^2)*f(x^3))/(f(x)*f(x)); Coefficients(F);`

- but based upon the previous code, it is hard to find the $n^{\text{th}}$-coefficient, and this one is a little bit better:
`PQ<x> := PowerSeriesRing(RationalField()); N :=30; f := func< x | &*[(1 - x^(i)) : i in [1..N]] >; (f(x^2)*f(x^3))/(f(x)*f(x));`

But this code is not interesting either, because for example to calculate the first $20$ coefficients, we need to consider $N$ large enough ($20 \leq N$), and then do the multiplication, and then search for the term $x^n$, and its coefficient.