Öz For any prime power $q$ and any $u = (x_1,\dots ,x_n),v = (y_1,\dots ,y_n)\in \mathbb {F} _{q^2}^n$ set $\langle u,v\rangle := \sum _{i=1}^{n} x_i^qy_i$. For any $k\in \mathbb {F} _q$ and any $n\times n$ matrix $M$ over $\mathbb {F} _{q^2}$, the $k$-numerical range $\mathrm{Num} _k(M)$ of $M$ is the set of all $\langle u,Mu\rangle$ for $u\in \mathbb {F} _{q^2}^n$ with $\langle u,u\rangle =k$ \cite{cjklr}. Here, we study the case $q=2$, which is quite different from the case $q\ne 2$.