Modular forms and ellipsoidal T-designs

In recent work, Miezaki introduced the notion of a spherical T-design 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.