After reading many textbooks I still can't get the jargon correct. Given a spherical harmonic $u \in L^2(S^n)$ one could construct a theta function:

$$ \theta (z;u) = \sum_{ m \in \mathbb{Z}^3} u (m) e^{2\pi i \, |m|^2}$$

If $u$ is not constant. This thing is a $\Gamma_0(4)$ cusp form. Could these theta functions be automorphic forms?

After reading many textbooks I still can't get the jargon correct. Given a spherical harmonic $u \in L^2(S^n)$ one could construct a theta function:

$$ \theta (z;u) = \sum_{ m \in \mathbb{Z}^3} u (m) e^{2\pi i \, |m|^2\,z}$$

with $z\in \mathfrak H$. The situation of interest is that $u$ is not constant. This thing is a $\Gamma_0(4)$ cusp form. Could these theta functions be automorphic forms?