Tetragonal modular quotients $X_0^{+}(N)$

In this paper we determine all quotient curves $X_0^{+}(N)$ whose $\mathbb{Q}$ or $\mathbb{C}$-gonality is equal to $4$. As a consequence, we find several new cases when the modular curve $X_0(N)$ has $\mathbb{Q}$-gonality equal to $8$.