Let $f$ be a modular form for $\Gamma$ a congruence subgroup of $\sldeuxz$ . \begin{equation} f(z)=\somme{n=0}{\infty}a_{n}q^n \end{equation}where $q=e^{2i\pi \tau}$ .

For $m\in \mathbb N$ , let $T_{m}f(z)=\somme{n=0}{\infty}b_{n}q^n$ with : \begin{equation} b_{n}=\somme{d|\gcd(m,n)}{}d^{k-1}a_{mn/d^2} \end{equation}In particular, for $p$ a prime, $T_{p}f(z)=\somme{n=0}{\infty}b_{n}q^n$ with; \begin{equation} b_{n}=a_{pn}+p^{k-1}a_{n/p} \end{equation}where $a_{n/p}=0$ if $n$ is not divisible by $p$ .

The operator $T_n$ is a linear operator on the space of modular forms called a Hecke operator.

The Hecke operators leave the space of modular forms and cusp forms invariant and turn out to be self-adjoint for a scalar product called the Petersson scalar product. In particular they have real eigenvalues. Hecke operators also satisfy multiplicative properties that are best summarized by the formal identity: \begin{equation} \underset{n=1}{\overset{\infty}{\sum}}T_{n}n^{-s}=\underset{p}{\prod}(1- T_{p}p^{-s}+p^{k-1-2s})^{-1} \end{equation}That equation in particular implies that $T_mT_n=T_nT_m$ whenever $\gcd(n,m)=1$ .

The set of all Hecke operators is usually denoted $\mathbb T$ and is called the Hecke algebra.

Group algebra example

Such $\mathcal{H}(G)$ algebras play an important role in the theory of decomposition of group representations into tensor products.