Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane.
Question. Can one show that the unit circle lies in $\lambda(\{x+iy: -1\leq x < 1 , y>1/2\})?$