# §23.18 Modular Transformations

## Elliptic Modular Function

$\mathop{\lambda\/}\nolimits\!\left(\mathcal{A}\tau\right)$ equals

 23.18.1 $\mathop{\lambda\/}\nolimits\!\left(\tau\right),$ $1-\mathop{\lambda\/}\nolimits\!\left(\tau\right),$ $\frac{1}{\mathop{\lambda\/}\nolimits\!\left(\tau\right)},$ $\frac{1}{1-\mathop{\lambda\/}\nolimits\!\left(\tau\right)},$ $\frac{\mathop{\lambda\/}\nolimits\!\left(\tau\right)}{\mathop{\lambda\/}% \nolimits\!\left(\tau\right)-1},$ $1-\frac{1}{\mathop{\lambda\/}\nolimits\!\left(\tau\right)},$ Symbols: $\mathop{\lambda\/}\nolimits\!\left(\NVar{\tau}\right)$: elliptic modular function and $\tau$: complex variable Permalink: http://dlmf.nist.gov/23.18.E1 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for 23.18

according as the elements $\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ of $\mathcal{A}$ in (23.15.3) have the respective forms

 23.18.2 $\begin{bmatrix}\mathrm{o}&\mathrm{e}\\ \mathrm{e}&\mathrm{o}\end{bmatrix},$ $\begin{bmatrix}\mathrm{e}&\mathrm{o}\\ \mathrm{o}&\mathrm{e}\end{bmatrix},$ $\begin{bmatrix}\mathrm{o}&\mathrm{e}\\ \mathrm{o}&\mathrm{o}\end{bmatrix},$ $\begin{bmatrix}\mathrm{e}&\mathrm{o}\\ \mathrm{o}&\mathrm{o}\end{bmatrix},$ $\begin{bmatrix}\mathrm{o}&\mathrm{o}\\ \mathrm{e}&\mathrm{o}\end{bmatrix},$ $\begin{bmatrix}\mathrm{o}&\mathrm{o}\\ \mathrm{o}&\mathrm{e}\end{bmatrix}.$ Symbols: e: even integers and o: odd integers Permalink: http://dlmf.nist.gov/23.18.E2 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for 23.18

Here e and o are generic symbols for even and odd integers, respectively. In particular, if $a-1,b,c$, and $d-1$ are all even, then

 23.18.3 $\mathop{\lambda\/}\nolimits\!\left(\mathcal{A}\tau\right)=\mathop{\lambda\/}% \nolimits\!\left(\tau\right),$ Symbols: $\mathop{\lambda\/}\nolimits\!\left(\NVar{\tau}\right)$: elliptic modular function, $\tau$: complex variable and $\mathcal{A}$: bilinear transformation Permalink: http://dlmf.nist.gov/23.18.E3 Encodings: TeX, pMML, png See also: Annotations for 23.18

and $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ is a cusp form of level zero for the corresponding subgroup of SL$(2,\mathbb{Z})$.

## Klein’s Complete Invariant

 23.18.4 $\mathop{J\/}\nolimits\!\left(\mathcal{A}\tau\right)=\mathop{J\/}\nolimits\!% \left(\tau\right).$ Symbols: $\mathop{J\/}\nolimits\!\left(\NVar{\tau}\right)$: Klein’s complete invariant, $\tau$: complex variable and $\mathcal{A}$: bilinear transformation Referenced by: §23.18 Permalink: http://dlmf.nist.gov/23.18.E4 Encodings: TeX, pMML, png See also: Annotations for 23.18

$\mathop{J\/}\nolimits\!\left(\tau\right)$ is a modular form of level zero for SL$(2,\mathbb{Z})$.

## Dedekind’s Eta Function

 23.18.5 $\mathop{\eta\/}\nolimits\!\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A}% )\left(-i(c\tau+d)\right)^{1/2}\mathop{\eta\/}\nolimits\!\left(\tau\right),$

where the square root has its principal value and

 23.18.6 $\varepsilon(\mathcal{A})=\mathop{\exp\/}\nolimits\!\left(\pi i\left(\frac{a+d}% {12c}+s(-d,c)\right)\right),$
 23.18.7 ${s(d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}% \right\rfloor-\frac{1}{2}\right),}$ $c>0$. Symbols: $\left\lfloor\NVar{x}\right\rfloor$: floor of $x$, $\left(\NVar{m},\NVar{n}\right)$: greatest common divisor (gcd), $c$: integer and $d$: integer Referenced by: §23.18, Equation (23.18.7) Permalink: http://dlmf.nist.gov/23.18.E7 Encodings: TeX, pMML, png Errata (effective with 1.0.11): Originally the sum $\sum_{r=1}^{c-1}$ was written with an additional constraint on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed. Reported 2015-10-05 by Howard Cohl and Tanay Wakhare See also: Annotations for 23.18

Here $s(d,c)$ is a Dedekind sum. See (27.14.11), §27.14(iii), §27.14(iv) and Apostol (1990, pp. 48 and 51–53). Note that $\mathop{\eta\/}\nolimits\!\left(\tau\right)$ is of level $\tfrac{1}{2}$.