Integral echelon basis of level-one modular forms
= Integral echelon basis of level-one modular forms
If $N+1=\dim M_k(SL_2(\mathbb Z))$, there is a basis $f_0,\ldots,f_N$ with integral Fourier coefficients and
$$
f_i=q^i+\sum_{n\geq N+1}a_n(f_i)q^n.
$$
Start from the integral forms $\Delta^iE_4^aE_6^b$ and use integer elimination on their unitriangular leading coefficients.