Prime-detecting quasimodular forms lie in the Eisenstein space

Prove that every quasimodular form whose Fourier coefficients are nonnegative for all positive indices and vanish if and only if the index n ≥ 2 is prime belongs to the quasimodular Eisenstein space; equivalently, establish that the set Ω of such prime-detecting quasimodular forms equals E ∩ Ω, where E is the additive span of even-weight Eisenstein series and their derivatives.

Background

The paper studies quasimodular forms whose Fourier coefficients detect primes (nonnegative at all positive indices and zero precisely at primes). Let E denote the quasimodular Eisenstein space (additive span of even-weight Eisenstein series and their derivatives) and Ω denote the set of prime-detecting quasimodular forms.

Motivated by numerical evidence and the structural results in Section 2 (particularly Theorem 2.3 describing E ∩ Ω), the authors conjecture that any prime-detecting quasimodular form must in fact lie in the Eisenstein subspace, i.e., no additional prime-detecting forms exist outside E.