Modular Forms for $\mathrm{G}\mathrm{L}(r,{\mathbb{F}}_q[T])$: $t$-expansions of the basic forms

We give closed formulas for the first few expansion coefficients of the basic modular forms for $\mathrm{G}\mathrm{L}(r,{\mathbb{F}}_q[T])$. Here the rank $r$ is larger or equal to $3$, and the forms in question include the coefficient forms $g_1,\dots ,g_r$ and the Eisenstein series $E_{q^i-1}$ ($i\in \mathbb{N}$).