Quasimodular forms that detect primes are Eisenstein

Abstract: MacMahon's partition functions and their extensions provide equations that identify prime numbers as solutions. These results depend on the theory of (mixed weight) quasimodular forms on $SL_2(\mathbb{Z})$. Two of the authors, along with Craig, conjectured an explicit description of the set of prime-detecting quasimodular forms in terms of Eisenstein series and their derivatives. Kane et al.\ recently verified this conjecture using analytic methods. We offer an alternative proof using the theory of $\ell$-adic Galois representations associated to modular forms.