I'm trying to understand an argument of Deligne (in Courbes elliptiques: Formulaire d'après J. Tate), but I'm not familiar enough with algebraic geometry, so I'm getting quite confused. So even in my statement I might say things that don't make any sense, please correct me if so.

An elliptic curve is a smooth proper morphism $p : E \rightarrow S$ of schemes, such that the geometric fibers are connected curves of genus 1, along with a section $e: S \rightarrow E$. The sheaf of differentials $\Omega_{E/S}$ can be pushed forward to get a sheaf $\omega_{E/S} = p_{*}\Omega_{E/S}$ on $S$.

Next, he shows how $E$ can be embedded in projective space $\mathbb{P}_S^2$, in non-homogeneous coordinates: $ (*)\space\space\space\space y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$.

Then later on, he defines modular forms of weight $n$ as being a law that associates to each elliptic curve $E \rightarrow S$ as above, a section of $\omega_{E/S}^{\otimes n}$ in a manner which is compatible with base change. He next says: applying the definition to the equation above (*), we see that any modular form of weight $n$ is a polynomial of degree $n$ in the $a_i$'s. I sort of see how that might make sense, but I'd appreciate it if someone could explain a bit more to me how to explicitly derive that conclusion (that they're polynomials in the $a_i$'s). Thanks in advance!

Edit: extra question: for a general scheme $S$, what do the $a_i$'s mean? if $S = Spec(R)$, I expect them to be elements of $R$, but what are they in the case of a more complicated scheme?