Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 3 / b / Solution

The Klein j-invariantis a weight-zero modular function for $\Gamma (1)$, holomorphic on $\mathfrak{h}$ and meromorphic at infinity. LetSince $c$ is even,The modular invariance of $j$ therefore givesThe quotient is meromorphic on $\mathfrak{h}$ and at the cusps, so it is a weight-zero modular function of level ${\Gamma }_1(2)$. This is the Level-two modular ratio of Klein j-invariants.