Modular forms for \,GL\,(r, Fq[T]): t-expansions of the basic forms

We give closed formulas for the first few expansion coefficients of the basic modular forms for \,GL\,(r, Fq[T]) GL ( r , F q [ T ] ) . Here the rank r r is larger or equal to $$3$$ 3 , and the forms in question include the coefficient forms g₁, , gᵣ g 1 , ⋯ , g r and the Eisenstein series E_qⁱ-1 E q i - 1 ( i ∈ N i ∈ N ).