Modular forms and ellipsoidal T-designs

In recent work, Miezaki introduced the notion of a $spherical$ $T$-d$esign$ in $\mathbb{R}^2$, where $T$ is a potentially infinite set. As an example, he offered the $\mathbb{Z}^2$-lattice points with fixed integer norm (a.k.a. shells). These shells are $maximal$ spherical $T$-designs, where $T=\mathbb{Z}^+\setminus 4\mathbb{Z}^+$. We generalize the notion of a spherical $T$-design to special ellipses, and extend Miezaki's work to the norm form shells for rings of integers of imaginary quadratic fields with class number 1.