= Integral triangular basis of level-one modular forms
{title2=$F_j=\Delta^jE_4^{a_j}E_6^{b_j}=q^j+O(q^{j+1})$}
Choose $4a_j+6b_j=k-12j$ with nonnegative exponents, for every $j$ with $k-12j\ge0$ and $k-12j\ne2$. 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 $E_4,E_6$.
Back to article page
