Use that the complex vector space of weight-$4$ modular forms on $\mathrm{SL}_2(\mathbb{Z})$ is the one-dimensional space $\mathbb{C} \cdot E_4$.

@FedorPetrov: Yes, exactly. Just because it should presumably be a smoother function of $n$ (as for the cyclotomic case). By the way, the irreducibility of these polynomials $G_n$ is conjectured but not proved: they should be the irreducible factorization of the iteration.