A fundamental region for Hecke's modular group

Hecke proved analytically that when λ ≥ 2 or when $\text{λ = 2}\text{cos}\text{(}\text{π}\text{q}\text{)}$, q ∈ Z, q ≥ 3, then $\text{B(λ) = {τ:}\text{Im}\text{τ > 0, |}\text{Re}\text{τ| <}\text{λ}\text{2}\text{, |τ| > 1}}$ is a fundamental region for the group G(λ) = 〈S_λ, T〉, where S_λ: τ → τ + λ and $\text{T: τ →}\text{?1}\text{τ}$. He also showed that B(λ) fails to be a fundamental region for all other λ > 0 by proving that G(λ) is not discontinuous. We give an elementary proof of these facts and prove a related result concerning the distribution of G(λ)-equivalent points.