just a disclaimer before my question, I don't really know too much about modular forms, as I just started learning it. However, any answers whether technical or not are very welcomed!
I know that modular forms are useful in quadratic form theory. Some examples I know of are counting the number of representations of an integer as sum of four integers ($s_4(n)$), and Tunnell's theorem on congruent numbers. My main question is whether the theory is applicable to higher powers or not, e.g. counting $S(n) = \left\{a^4 + b^4 + c^4 + d^4 : (a, b, c, d) \in \mathbb{Z}^4\right\}$. I suppose that if it can be, then one would form the generating function $\Theta_4(q) := \sum_{k \in \mathbb{Z}} q^{k^4}$ and analyse that. However, the proof of $\Theta_2^4$ satisfying the transformation formula for modular forms, which AFAIK among many other things, uses the fact that $f(x) = e^{-\pi x^2}$ is the fourier transform of itself, and I don't think any analogous results hold for $\Theta_4$. Hence, I suppose that the theory can't be directly applied this way, but I guess my real question is:
Are there any other possible approaches to still use modular forms or generalisations of it to attack this problem, or is it really limited to quadratic forms? Are there any "intuitive" reasons to why that is?
(Other analytic NT methods like circle method is irrelevant here.)