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I already asked a closely related question on MO, but didn't received any answer.

Considering the modular lambda function, the values of $ \lambda^{*}(n) $ for some integers are given on here.

Is there a way to calculate the coefficients of the minimal polynomials of each $ \lambda^{*}(n) $ ?

Here you have the minimal polynomials for the first $10$ positive integers

\begin{array}{|c|c|} \hline n & P_n(x) \\ \hline 1 & 2x^2-1 \\ \hline 2 & x^2+2x-1 \\ \hline 3 & 16x^4-16x^2+1 \\ \hline 4 & x^2-6x+1 \\ \hline 5 & 16x^{8} - 32x^{6} + 88x^4 - 72 x^2 + 1 \\ \hline 6 & x^{4} + 12x^{3} +2x^2 - 12x +1 \\ \hline 7 & 256x^4 - 256x^2 + 1 \\ \hline 8 & x^{4} - 20x^{3} - 26x^2 - 20x +1 \\ \hline 9 & 16x^{8} - 32x^{6} + 792x^4 - 776x^2 + 1 \\ \hline 10 & x^{4} + 36x^{3} + 2x^2 - 36 x + 1 \\ \hline \end{array}

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  • $\begingroup$ Mathematica code is given in oeis.org/A084540 $\endgroup$
    – Max Alekseyev
    Commented Apr 29 at 11:54
  • $\begingroup$ @MaxAlekseyev yes I know, I used that code to find the polynomials, but I wonder if it's possible to compute the coefficients analytically $\endgroup$
    – user967210
    Commented Apr 29 at 15:09

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