 Hecke operator on lattice functions

One standard normalization isWith lattice model of a modular form homogeneity $F(uL)=u^{-k}F(L)$, the factor $1/n$ gives the usual weight-$k$ Hecke operator. For prime $p$, the overlattices contribute $f((\tau +b)/p)$ and $p^{k}f(p\tau )$; division by $p$ produces the standard Fourier coefficients.