I'm reading Zagier's paper "Quantum Modular Forms", and there is a part that I can't understand well.
In his paper, the example 1 (which are closely related to Maass wave form) said that the function $$ f(x)=q^{1/24}(1+\sum_{n\geq 0}(-1)^{n}(q;q)_{n}q^{n+1}),\qquad q=e^{2\pi i x}, x\in \mathbb{Q} $$ is a quantum modular form of weight 1 with character, i.e. $$ f(x+1)=\zeta_{24}f(x), \qquad\frac{1}{2x+1}f\left(\frac{x}{2x+1}\right)=\zeta_{24}f(x)+h(x) $$ where $h$ can be extended smoothly to $\mathbb{R}$ except $x=-1/2$. However, I tried to draw graph by using MATLAB of $h$ and I got slightly different graph from the author's :
Two graphs coincide for $x>-1/2$ but not for $x<-1/2$. (Mine represents imaginary part of $h$.)
Actually, the character $\chi:\Gamma_{0}(2)\to \mathbb{C}^{\times}$ defined by $$ \chi(T)=\chi\left(\begin{matrix}1&1\\0&1\end{matrix}\right)=\zeta_{24}, \chi(R)=\chi\left(\begin{matrix}1&0\\2&1\end{matrix}\right)=\zeta_{24} $$ should satisfy $\chi(-I)=\chi((RT^{-1})^{2})=1$. But $f$ has odd weight (weight 1), so it would be natural to have $\chi(-I)=-1$. There might be something I missed. Thanks in advance.