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 $1$ 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 $\ell$-adic Galois representations, we prove that any prime-detecting quasimodular form on $\Gamma_{0}(N)$ belongs to the direct sum of the spaces of quasimodular Eisenstein series and quasimodular oldforms. Moreover, for a quasimodular form $f$ that is not prime-detecting, we give an upper bound for the number of primes $p$ less than $X$ for which the $p$-th Fourier coefficient of a quasimodular form vanishes.