Given a quadratic form Q in k variables, there is an associated theta series $$\theta_Q(z) = \sum_{x\in \mathbb{Z}^k}q^{Q(x)}$$ where $q = e^{2\pi i z}$ which is a modular form of weight $k/2$. Thus the coefficient on $q^n$ counts the number of solutions in $\mathbb{Z}^k$ to $Q(x) = n$; denote this quantity $r_Q(n)$. The level of these modular forms is related to the level of their corresponding quadratic forms. Since these modular forms live inside finite-dimensional vector spaces, one question to ask is whether there are any linear relationships among them. At least among the ternary forms for which I've looked at this computationally, there often is a great deal of linear dependence among forms of this type that live in the same space, and so there are a lot of linear relationships among the $r_Q(n)$ for various $Q$.

My question is if there are "other reasons" (i.e. not related to modular considerations) to expect that the $r_Q(n)$ for various $Q$ should be related in this fashion. Is there a reason *a priori* to expect that these theta series should be heavily linearly-dependent on one another? Is there a more combinatorial approach (or an alternative number-theoretic approach) to counting solutions to quadratic forms that could suggest the form that these relationships might take? I'm asking because in the course of my computations with ternary forms I observed that twisting forms by quadratic characters always (at least for all the examples I computed) gave a form that was linearly dependent on untwisted forms that lived in the same space. Is there a reason to suspect why this should always be the case?