Let $f$ be a holomorphic modular form of weight $k$, level $N$, and character $\chi$, over the group $\Gamma_0(N)$.

Let $n$ be a positive integer. Let $(\beta_i)_{1\leq i\leq s}$ be a system of representatives of the left action of $\Gamma_0(N)$ on $\Gamma_n(N)$, with $\beta_i=\left(\begin{smallmatrix}a_i&b_i\\c_i&d_i\end{smallmatrix}\right)$. We define the **Hecke operator of index $n$** acting on $f$ to be:
\[
T(n)f=n^{k/2-1}\sum_{1\leq i\leq s}\chi(a_i)f|_k\beta_i,
\]
where $f|_k\beta_i$ is the weight $k$ slash-action.

Note that this definition is independent of the choice of coset representatives $\beta_i$ for the cosets $\Gamma_0(N)\setminus\Gamma_n(N)$.

The Hecke operators are **$\chi$-hermitian** with respect to the Petersson scalar product, i.e. if $f$ and $g$ are in $M_k(\Gamma_0(N),\chi)$, at least one of them a cusp form and $n$ is coprime to $N$, then
\[
\langle T(n)f,g\rangle_{\Gamma_0(N)}=\chi(n)\langle f,T(n)g\rangle_{\Gamma_0(N)}
\]

The Hecke operators **preserve** modular forms and cusp forms.

They form an **algebra** which is a commutative ring.

**Authors:**