 Marked-lattice model of a modular form

Choose an oriented lattice basis with $t={\omega }_2/N$ modulo the lattice. Changing such a basis is exactly the Gamma 1 congruence subgroup action, so the weight law makes the displayed function independent of the choice. It satisfies $F(uL,ut)=u^{-k}F(L,t)$. Holomorphy on the half-plane and at all modular cusp degenerations identifies the functions coming from modular forms.