Lattice model of a modular form ($V_k$)

Let $\mathcal{L}$ be the set of complex lattices. A weight-$k$ lattice function satisfies $F(\lambda \Lambda )={\lambda }^{-k}F(\Lambda )$. Evaluating at ${\Lambda }_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$ identifies such functions with weight-$k$ modular-invariant functions on the upper half-plane.

One standard normalization isUnder the lattice model this becomes the usual weight-$k$ Hecke operator.