Numerical ranges over finite fields

We explore an arithmetic analogue of the numerical range. We define the numerical range of a square matrix with entries in a finite field Zp[i]Zp[i], for any prime p   congruent to 3 modulo 4. We establish the basic properties of these new numerical ranges, and prove several foundational results for matrices of arbitrary dimension. We classify the shapes of the numerical ranges of 2×22×2 matrices over these finite fields.