In his notebooks Ramanujan mentions something called a "complete series" which is some power series $\sum_{n = 0}^{\infty}a_{n}q^{n}$ in terms of $q = e^{-y}$ with $y = \pi K'/K$ and $z = 2K/\pi$ such that the expression $$S = \frac{1}{z^{p}}\sum_{n = 0}^{\infty}a_{n}q^{n}$$ is an algebraic function of $k = \vartheta_{2}^{2}(q)/\vartheta_{3}^{2}(q)$

The exact quote of Ramanujan is given in Entry 1, Chap 21, Ramanujan's Notebooks vol 3 by Bruce C. Berndt. Here Ramanujan just says that the value $S$ defined above can be expressed in terms of radicals.

Then Ramanujan gives examples of Eisenstein Series $E_{4}$ and $E_{6}$ defined as $$E_{4}(q) = 1 + 240\sum_{n = 1}^{\infty}\frac{n^{3}q^{n}}{1 - q^{n}}, E_{6}(q) = 1 - 504\sum_{n = 1}^{\infty}\frac{n^{5}q^{n}}{1 - q^{n}}$$ Indeed with factors $1/z^{4}$ and $1/z^{6}$ we can see that $E_{4}/z^{4}$ and $E_{6}/z^{6}$ are in fact polynomials in $k = \vartheta_{2}^{2}(q)/\vartheta_{3}^{2}(q)$.

I believe that Ramanujan did have his own sense of a modular form and he called it a "complete series". Going by his ideas let's formulate the following definition.

*Let $0 < q < 1$ and let $k = \vartheta_{2}^{2}(q)/\vartheta_{3}^{2}(q)$ and $z = \vartheta_{3}^{2}(q)$. A function $f(q)$ defined by a power series $f(q) = \sum_{n = 0}^{\infty}a_{n}q^{n}$ is said to be modular form of weight $p > 0$ if the expression $f(q)/z^{p}$ is an algebraic function of $k$.*

By using the power series we just want to make sure that $f(q)$ is analytic in $|q| < 1$. I want to know whether such a formulation is acceptable (at least to some extent)? How far does this match with the modern definition of a "modular form"?