We determine the action of the Hecke operators \(T_{\mathfrak {p},i}\) on the coefficient forms \(g_{1}, \ldots , g_{r-1}, g_{r} = \Delta \) , and h, which together generate the ring of modular forms for \({{\,\textrm{GL}\,}}(r, \mathbb {F}_{q}[T])\) . All these are eigenforms with powers of \(\pi \) as eigenvalues, where \(\pi \) is the monic generator of the prime ideal \(\mathfrak {p}\) of \(\mathbb {F}_{q}[T]\) . We further describe the growth of the t-expansion coefficients of the discriminant function \(\Delta \) . It is such that the product expansion of \(\Delta \) as well as the t-expansion of each modular form converges on the natural fundamental domain for \({{\,\textrm{GL}\,}}(r, \mathbb {F}_{q}[T])\) .
