On the numerical range of matrices over a finite field

Let q be a prime power. Following a paper by Coons, Jenkins, Knowles, Luke and Rault (case q   a prime p≡3(mod4)) we define the numerical range Num(M)⊆Fq2Num(M)⊆Fq2 of an n×nn×n-matrix M   with coefficients in Fq2Fq2 in terms of the usual Hermitian form. We prove that ♯(Num(M))>q♯(Num(M))>q (case q≠2q≠2), unless M   is unitarily equivalent to a diagonal matrix with eigenvalues contained in an affine FqFq-line. We study in details Num(M)Num(M) when n=2n=2.