Hecke operator

Summary

The Hecke operators $T_n$, for $n \ge 1$, are linear maps on the space $M_k$ of modular forms of weight $k$. They average a modular form over the sublattices of index $n$ of a lattice. They commute with each other and preserve the cusp forms. A common eigenform of all $T_n$, suitably normalized, has multiplicative Fourier coefficients and an L-function with an Euler product.