We study properties of the numerical range of the Foguel–Halmos operator FT = [ S0∗ TS] on ℓ2 ⊕ ℓ2, where S is the simple unilateral shift and T = diag(a1, a2, . . .) with an = 1 if n = 3k for some k ≥ 1 and an = 0 otherwise. Among other things, we show that the numerical range W(FT ) is neither open nor closed, and give lower and upper bounds for the numerical radius w(FT ).