On the structure of prime-detecting quasimodular forms in higher levels

Craig, van Ittersum, and Ono conjectured that every prime-detecting quasimodular form of level [Formula: see text] is a quasimodular Eisenstein series. This conjecture was proved by Kane–Krishnamoorthy–Lau and by van Ittersum–Mauth–Ono–Singh independently. However, in higher levels, prime-detecting quasimodular forms need not be Eisenstein. Recently, Kane, Krishnamoorthy, and Lau formulated a natural higher level analogue of the above conjecture and proved it by analytic methods. In a similar direction, but via an alternative approach based on the independence of characters of [Formula: see text]-adic Galois representations, we prove that any prime-detecting quasimodular form on [Formula: see text] belongs to the direct sum of the spaces of quasimodular Eisenstein series and quasimodular oldforms. Moreover, for a quasimodular form [Formula: see text] that is not prime-detecting, we give an upper bound for the number of primes [Formula: see text] less than [Formula: see text] for which the [Formula: see text]th Fourier coefficient of a quasimodular form vanishes.