An automorphic factor of weight k is a function

An automorphic factor must satisfy:

1. For a fixed ${\displaystyle \gamma \in \Gamma }$ , the function ${\displaystyle \nu (\gamma ,z)}$  is a holomorphic function of ${\displaystyle z\in \mathbb {H} }$ .
2. For all ${\displaystyle z\in \mathbb {H} }$  and ${\displaystyle \gamma \in \Gamma }$ , one has
   $${\displaystyle \vert \nu (\gamma ,z)\vert =\vert cz+d\vert ^{k}}$$

    
   for a fixed real number k.
3. For all ${\displaystyle z\in \mathbb {H} }$  and ${\displaystyle \gamma ,\delta \in \Gamma }$ , one has
   $${\displaystyle \nu (\gamma \delta ,z)=\nu (\gamma ,\delta z)\nu (\delta ,z)}$$

    
   Here, ${\displaystyle \delta z}$  is the fractional linear transform of ${\displaystyle z}$  by ${\displaystyle \delta }$ .
4. If ${\displaystyle -I\in \Gamma }$ , then for all ${\displaystyle z\in \mathbb {H} }$  and ${\displaystyle \gamma \in \Gamma }$ , one has
   $${\displaystyle \nu (-\gamma ,z)=\nu (\gamma ,z)}$$

    
   Here, I denotes the identity matrix.

Every automorphic factor may be written as

${\displaystyle \nu (\gamma ,z)=\upsilon (\gamma )(cz+d)^{k}}$ 

with

${\displaystyle \vert \upsilon (\gamma )\vert =1}$ 

The function ${\displaystyle \upsilon :\Gamma \to S^{1}}$  is called a multiplier system. Clearly,

${\displaystyle \upsilon (I)=1}$ ,

while, if ${\displaystyle -I\in \Gamma }$ , then

${\displaystyle \upsilon (-I)=e^{-i\pi k}}$ 

which equals ${\displaystyle (-1)^{k}}$  when k is an integer.

• Robert Rankin, Modular Forms and Functions, (1977) Cambridge University Press ISBN 0-521-21212-X. (Chapter 3 is entirely devoted to automorphic factors for the modular group.)