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A main possibly non-intuitive usage of "form" is as a somewhat particular type of map/function. Traditionally, the word function was used in a more restrained way and it was mainly used for real and complex functions, only. For example, classically in (real) functional analysis one would have: A function would be a map from $\mathbb{R}$ to $\mathbb{R}... 2 Here is the general picture. Let$f \in \mathcal{M}_k(N,\chi)$be a modular form of weight$k$, level$N$, and nebentypus$\chi$. Suppose that$f$has the Fourier expansion $f(z) = \sum_{m = 0}^{\infty} a_f(m) e(mz).$ Then by replacing$z$with$nz$, we see that $f(nz) = \sum_{m = 0}^{\infty} a_f(m) e(mnz) = \sum_{m = 0}^{\infty} a_{f_n}(m) e(mz)$ with \[... 1 I'm not a Mathistorian, but... Likely it originally meant its English meaning of "appearance", and it still does in most usages. Quadratic forms have a very specific appearance, namely a homogenous quadratic polynomial. Modular forms are functions satisfying a certain form of equation and some other conditions. Conjunctive/disjunctive/Skolem normal forms are ... 1 So I'm not sure about references specific to Hauptmoduln (except specialized research articles), so perhaps you would be better served by reading up on modular functions, forms and curves. I suggest Milne's online notes which are excellent, or Diamond and Shurmon's book which is also excellent. As far as your specific and interesting question regarding the ... 1 I think the argument is something like this. Note that$t=t(\tau)$is a meromorphic modular function for$\Gamma_1(6)$. Therefore, if$E(t)(f(t)−\zeta(3))$is a polynomial in$t$, then$E(\tau)(f(\tau)-\zeta(3))$would be a meromorphic modular function for$\Gamma_1(6)$. As you said,$E(\tau)$is a modular form of weight$2$for$\Gamma_1(6)\$. It follows ...