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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.

Let $\mathcal{L}$ be the set of lattices in $\mathbb{C}$. The lattice model of a modular form isFor the normalization used here, the $n$th Hecke operator on lattice functions isThere are finitely many such superlattices, and scaling gives a bijection between those above $\Lambda$ and those above $\lambda \Lambda$, so $T_{n}F$ again has weight $k$.