Klein j-invariant ($j(\tau )$)

The Klein j-invariant is the weight-zero level-one modular functionIt is holomorphic on the upper half-plane and has a simple pole at infinity.

The function $j(\tau )/j(2\tau )$ has weight zero and level ${\Gamma }_1(2)={\Gamma }_0(2)$. It has a simple zero at the cusp at infinity and a simple pole at the cusp zero.