  A modular grid is a pair of sequences $(f_m)_m$ and $(g_n)_n$ of weakly
holomorphic modular forms such that for almost all $m$ and $n$, the coefficient
of $q^n$ in $f_m$ is the negative of the coefficient of $q^m$ in $g_n$. Zagier
proved this coefficient duality in weights $1/2$ and $3/2$ in the Kohnen plus
space, and such grids have appeared for Poincar\'{e} series, for modular forms
of integral weight, and in many other situations. We give a general proof of
coefficient duality for canonical row-reduced bases of spaces of weakly
holomorphic modular forms of integral or half-integral weight for every group
$\Gamma \subseteq {\text{SL}}_2(\mathbb{R})$ commensurable with
${\text{SL}}_2(\mathbb{Z})$. We construct bivariate generate functions that
encode these modular forms, and study linear operations on the resulting
modular grids.
