I already asked the same question here, but received no answer. I did some little progress and so I'm asking again.
I was playing with the theta functions with argument $ z = 0 $
$ \vartheta_2(q) =\sum_{n=-\infty}^\infty q^{(n+1/2)^2} $
$ \vartheta_3(q) =\sum_{n=-\infty}^\infty q^{n^2} $
$ \vartheta_4(q) =\sum_{n=-\infty}^\infty (-1)^nq^{n^2} $
And I noticed that the ratio $ \frac{\vartheta_4(e^{-2\pi})}{\vartheta_3(e^{-2\pi})}$ is the root of the irreducible polynomial $x^8+32x^4-32$, so I kept going and I found the same kind of relations between these ratios and irreducible polynomials .
As the user @Somos pointed out, the context is singular moduli, and defining$ \,q_n:=\exp(-\pi\sqrt{n})\, $ and the irreducible polynomials $ P_n(x) $, $ Q_n(x) $ and $ R_n(x) $, such that $ P_n(\vartheta_3(0,q_n)/\vartheta_2(0,q_n))=0\, $, $\,Q_n(\vartheta_4(0,q_n)/\vartheta_2(0,q_n))=0\, $ and $\,R_n(\vartheta_4(0,q_n)/\vartheta_3(0,q_n))=0\, $, we have this table
\begin{array}{|c|c|c|c|} \hline n & P_n(x) & Q_n(x) & R_n(x) \\ \hline 1 & x^4-2 & x-1 & 2x^4-1 \\ \hline 2 & x^4-2x^2-1 & x^8-4x^4-4 & x^8+4x^4-4 \\ \hline 3 & x^8-16x^4+16 & x^4-4x^2+1 & 16x^8-16x^4+1 \\ \hline 4 & x^2-2x-1 & x^8-32x^4-32 & x^8+32x^4-32 \\ \hline 5 & x^{16} - 72x^{12} + 88x^8 - 32x^4 + 16 & x^4 - 2x^3 - 2x^2 - 2x + 1 & 16x^{16} - 32x^{12} + 88x^8 - 72 x^4 + 1 \\ \hline 6 & x^8 - 12x^6 + 2x^4 + 12x^2 + 1 & x^{16} - 136x^{12} - 120x^8 + 32 x^4 + 16 & x^{16} + 136x^{12} - 120x^8 - 32 x^4 + 16 \\ \hline 7 & x^8 - 256 x^4 + 256 & x^4 - 16x^2 + 1 & 256x^8 - 256x^4 + 1 \\ \hline 8 & x^4 - 4 x^3 - 2 x^2 - 4 x + 1 & x^{16} - 448x^{12} - 1472x^8 - 2048x^4 - 1024 & x^{16} + 448x^{12} - 1472x^8 + 2048x^4 - 1024 \\ \hline 9 & x^{16} - 776x^{12} + 792x^8 - 32x^4 + 16 & x^4 - 4x^3 - 6x^2 - 4x + 1 & 16x^{16} - 32x^{12} + 792x^8 - 776x^4 + 1 \\ \hline 10 & x^8 - 36x^6 + 2x^4 + 36x^2 + 1 & x^{16} - 1288x^{12} - 1272 x^8 + 32 x^4 + 16 & x^{16} + 1288x^{12} - 1272x^8 - 32 x^4 + 16 \\ \hline \end{array}
As you can clearly notice, some patterns shows up: when $ n $ is odd $ P_n(x) $ and $ R_n(x) $ are reciprocal polynomials, while when $ n $ is even $ Q_n(x) $ and $ R_n(x) $ are equal except for the sign of the second and second-last coefficient.
These polynomials show up in other problems? Their coefficients could be found in any sequence?
