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In This MO question, Werner said that Hecke operator "changes" characters. I'm looking for any explicit theory of this kind, about Hecke operator with characters.

Actually, there are somewhat explicit example I found, for $\Gamma_{0}(4)$. Define $\chi:\Gamma_{0}(4)\to \mathbb{C}^{\times}$ as $\chi(T)=\chi(R)=e^{2\pi i/8}, \chi(-I)=1$, where $$ T=\begin{pmatrix}1&1\\0&1\end{pmatrix}, R=\begin{pmatrix}1&0\\4&1\end{pmatrix}, -I=\begin{pmatrix}-1&0\\0&-1\end{pmatrix} $$ which are generators of $\Gamma_{0}(4)$. Now let $f$ be a weight 0 modular form on $\Gamma_{0}(4)$ with character $\chi$, i.e. it satisfies $$ f(z+1)=e^{2\pi i /8}f(z), \,\,\,\,f\left(\frac{z}{4z+1}\right)=e^{2\pi i/8}f(z). $$ Now define $g(z)$ as $$ g(z)=f(3z)+f\left(\frac{z}{3}\right)+e^{10\pi i/8}f\left(\frac{z+1}{3}\right)+e^{4\pi i /8}f\left(\frac{z+2}{3}\right) $$ which is something looks like $T_{3}f$. Then with some tedious computations, I found that $$ g(z+1)=e^{6\pi i /8}g(z),\,\,\,\,g\left(\frac{z}{4z+1}\right)=e^{6\pi i /8}g(z) $$ holds. I just want to ask for any known results similar to this example.

Miyake's book about modular forms mentioned about Hecke operator with character and his Hecke operator does not changes character. But it assumes that the character can be extended to some bigger set multiplicatively, but the book doesn't give any necessary or sufficient conditions for this. For example, it is possible to define Hecke operator $T_{p}$ if we can extend the character $\chi:\Gamma_{0}(N)\to \mathbb{C}^{\times}$ to the bigger set $$\Gamma_{0}(N)\cup\Gamma_{0}(N)\begin{pmatrix}1&0\\0&p\end{pmatrix}\Gamma_{0}(N)$$ multiplicatively.

EDIT : To show the functional equation of $g$, I just computed a lot. The first one is not much complicated : we have \begin{align} g(z+1)&=f(3z+3)+f\left(\frac{z+1}{3}\right)+\zeta_{8}^{5}f\left(\frac{z+2}{3}\right)+\zeta_{8}^{2}f\left(\frac{z+3}{3}\right) \\ &=\zeta_{8}^{3}\left[f(3z)+f\left(\frac{z}{3}\right)+\zeta_{8}^{5}f\left(\frac{z+1}{3}\right)+\zeta_{8}^{2}f\left(\frac{z+2}{3}\right)\right] \\ &=\zeta_{8}^{3}g(z) \end{align} where $\zeta_{8}=e^{2\pi i /8}$. Second one is too long to write here..

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    $\begingroup$ If $f(z)$ is modular for $\Gamma,\chi$ then $f(3z)$ is modular for ${\scriptstyle\begin{pmatrix}1&0\\0&1/3\end{pmatrix}}\Gamma{\scriptstyle\begin{pmatrix}1&0\\0&3\end{pmatrix}}, \tilde{\chi}$ where $\tilde{\chi}(\gamma) = \chi({\scriptstyle\begin{pmatrix}1&0\\0&3\end{pmatrix}}\gamma{\scriptstyle\begin{pmatrix}1&0\\0&1/3\end{pmatrix}})$. How do you treat the 3 other terms ? $\endgroup$
    – reuns
    Commented Apr 17, 2017 at 2:01
  • $\begingroup$ @reuns I added computation. $\endgroup$
    – Seewoo Lee
    Commented Apr 17, 2017 at 3:36
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    $\begingroup$ Do you know if the kernel of $\chi$ is a congruence subgroup? (I suspect it probably is not, which might explain why Hecke operators behave so differently here from the case studied by Miyake.) $\endgroup$
    – David Loeffler
    Commented Apr 17, 2017 at 7:47
  • $\begingroup$ @DavidLoeffler I have no idea.. $\endgroup$
    – Seewoo Lee
    Commented Apr 17, 2017 at 9:02

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This was essentially done by Wohlfhart in his paper Über Operatoren Heckescher Art bei Modulformen reeller Dimension, and also explained in Stromberg's paper Hecke Operators for Maass Waveforms on $\mathrm{PSL}(2, \mathbb{Z})$ with Integer Weight and Eta Multiplier. One can also find the definition in my paper, with some application on quantum modular forms.

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