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.

A Gamma 1 level structure on a complex lattice $\Lambda$ is a point of exact order $N$ in $\mathbb{C}/\Lambda$. Similarity classes of such pairs are parametrized by ${\Gamma }_1(N)\backslash \mathfrak{h}$.

Multiplication of a marked point by a unit modulo $N$ gives the diamond action. In modular coordinates this is the slash action of a lift in ${\Gamma }_0(N)$ with lower-right entry $d$. Its Dirichlet character eigenspaces give nebentypus modular forms. For a nonzero form, the action of $-1$ imposes $\chi (-1)=(-1{)}^k$.

Sum over index-$p$ overlattices in which the marked point retains exact order $N$. There are $p+1$ summands at good primes and $p$ at bad primes. With homogeneity $F(uL,ut)=u^{-k}F(L,t)$, the coefficient $1/p$ is the normalization giving the standard Hecke operator Fourier action. Cusp holomorphy under rational slash operators proves preservation of modular forms.

At a prime not dividing the level, the root-of-unity average contributes $a_{pn}$ and the scaling overlattice contributes the second term. At a prime dividing the level that overlattice loses the marked point's exact order and is excluded. Extending the Dirichlet character by zero gives a uniform formula, with $a_{n/p}=0$ unless $p\mid n$. The bad-prime operator is $U_p$.

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.

A lattice has $p+1$ index-$p$ overlattices. All preserve the exact order of a Gamma 1 level structure of order $N$ when $p\nmid N$; exactly $p$ do so when $p\mid N$.

For an even nonnegative integer $k$,Poisson summation and the Fourier eigenfunction identity for $(x+iy{)}^k{e}^{-\pi (x^2+y^2)}$ give

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.