Let $A$ be an $d \times d$ matrix. The q-numerical range of $A$ is the set: \[ W_q(A) = \left\{ z \in \mathbb{C}: z = \bra{x}A\ket{y}, \braket{x}{y} = q, \ket{x} \in \mathbb{C}^d, \ket{y} \in \mathbb{C}^d, \braket{x}{x} = 1, \braket{y}{y} = 1 \right\}. \]
Properties of $W_q(A)$ of a matrix $A$ of dimension $d \times d$ [1], [2],[3]:
The numerical aproximation of $W_{13/14}(A)$, where $$ A = \begin{pmatrix} 0&1&1/2\\ 0&0&1\\ 0&0&0 \end{pmatrix}. $$
This example can be found in [5].