Let $A$ be an $d \times d$ matrix and $(c_1, c_2, \ldots, c_d)$ be a real d-tuple. The $C$-numerical range of $A$ is the set $$ W_C(A) = \left\{ \sum_{j=1}^d c_j \bra{x_j} A \ket{x_j}: \{\ket{x_i}\}_{i=1}^d \mathrm{\;forms\; an\; orthonormal\; basis\; of\;} \mathbb{C}^d \right\} $$