Integral triangular basis of level-one modular forms ($F_j={\Delta }^j{E}_4^{a_j}{E}_6^{b_j}=q^j+O(q^{j+1})$)

Choose $4a_j+6b_j=k-12j$ with nonnegative exponents, for every $j$ with $k-12j\ge 0$ and $k-12j\ne 2$. The Eisenstein series and the modular discriminant give modular forms with integer coefficients and distinct first powers of $q$. The valence formula for the modular group shows that these form a basis. Successively subtracting their leading terms proves that integral coefficients up to the greatest such $j$ force all coefficients to be integral. This is more precise than merely spanning by arbitrary monomials in $E_4,E_6$.