Relation between the elliptic Lambda function and j-invariant.

Question

Let $\lambda (= 16q - 128q^2 + 704q^3 - \cdots)$ be the elliptic Lambda function.

$j = \frac{256*(1 - \lambda + \lambda ^2)^3}{\lambda ^2 * (1 - \lambda)^2}$.

I calculated the following.

$j = \frac{256*(1 - 16q + 384q^2 - \cdots)^3}{(16q - 128q^2 + 704q^3 - \cdots)^2 * (1 - 16q + 128q^2 - 704q^3 - \cdots)^2} = \frac{1}{q^2} * \frac{(1 - 16q + 384q^2 - \cdots)^3}{(1 - 8q + 44q^2 - \cdots)^2 * (1 - 16q + 128q^2 - 704q^3 - \cdots)^2}$

$= \frac{1}{q^2} - 744 + 196884q^2 + 21493760q^4 + \cdots$.

$j = \frac{1}{q} - 744 + 196884q + 21493760q^2 + \cdots$, so I make a mistake. Where is wrong?

Answer 1

Let $\tau$ be in the upper half-plane. Some authors use $q$ to denote $\exp(2\pi i\tau)$. Others write $q$ for $\exp(\pi i\tau)$. This means that to translate between them, you sometimes have to replace $q$ by $q^2$ and vice versa.