Product of Eisenstein series with multiplicative power series

We say a power series $a_0+a_1q+a_2q^2+\cdots$ is \emph{multiplicative} if $n\mapsto a_n/a_1$ for positive integers $n$ is a multiplicative function. Given the Eisenstein series $E_{2k}(q)$, we consider formal multiplicative power series $g(q)$ such that the product $E_{2k}(q)g(q)$ is also multiplicative. For fixed $k$, this requirement leads to an infinite system of polynomial equations in the coefficients of $g(q)$. The initial coefficients can be analyzed using elimination theory. Using the theory of modular forms, we prove that each solution for the initial coefficients of $g(q)$ leads to one and only one solution for the whole power series, which is always a quasimodular form. In this way, we determine all solutions of the system for $k \le 20$. For general $k$, we can regard the system of polynomial equations as living over a symbolic ring. Although this system is beyond the reach of computer algebra packages, we can use a specialization argument to prove it is generically inconsistent. This is delicate because resultants commute with specialization only when the leading coefficients do not specialize to $0$. Using a Newton polygon argument, we are able to compute the relevant degrees and justify the claim that for $k$ sufficiently large, there are no solutions. These results support the conjecture that $E_{2k}(q)g(q)$ can be multiplicative only for $k = 2, 3, 4, 5, 7$.