On the detection of non-roots of D’Arcais polynomials

The Lehmer conjecture states that the non-constant Fourier coefficients of the 24th power of the Dedekind eta function are non-zero. Recently, Neuhauser and the first author exploited an easily accessible tool from algebraic number theory, namely the Dedekind–Kummer Theorem, to prove the non-vanishing of the Fourier coefficients of certain powers of the Dedekind eta function at roots of unity. We extend the application of this method to enlarge the scope of non-roots of the related D’Arcais polynomials.