On the numerical range of matrices defined over a finite field

Let q be a prime power. For u=(u1,…,un),v=(v1,…,vn)∈Fq2n let 〈u,v〉:=∑i=1nuiqvi be the Hermitian form of Fq2n. Fix an n×n matrix M over Fq2. Set Num(M):={〈u,Mu〉|u∈Fq2n,〈u,u〉=1} (the numerical range of M introduced by Coons, Jenkins, Knowles, Luke and Rault (case q a prime q≡3(mod4)) and by the author (arbitrary q)). When n=2 we prove an upper bound for |Num(M)|. We describe Num(M) for several classes of matrices, mostly for n=2,4.