# §23.18 Modular Transformations

## ¶ Elliptic Modular Function

equals

according as the elements of in (23.15.3) have the respective forms

23.18.2

Here e and o are generic symbols for even and odd integers, respectively. In particular, if , and are all even, then

and is a cusp form of level zero for the corresponding subgroup of SL.

## ¶ Klein’s Complete Invariant

is a modular form of level zero for SL.

## ¶ Dedekind’s Eta Function

where the square root has its principal value and

23.18.7.

Here the notation means that the sum is confined to those values of that are relatively prime to . See §27.14(iii) and Apostol (1990, pp. 48 and 51–53). Note that is of level .