 Solution

Let $N+1=\dim M_k(S{L}_2(\mathbb{Z}))$. For each $0\le i\le N$, the dimension formula, equivalently the valence formula for the modular group, ensures thatfor some nonnegative integers $a_i,b_i$. DefineThe modular discriminant, $E_4$, and $E_6$ have integral Fourier coefficients and leading terms $q$, $1$, and $1$, respectively. HenceTheir distinct orders of vanishing make the $h_i$ linearly independent, so they form a basis.

Starting with $f_N=h_N$, define $f_i$ downwards by subtracting from $h_i$ the integral multiples of $f_{i+1},\dots ,f_N$ needed to kill the coefficients of $q^{i+1},\dots ,q^N$. This integer Gaussian elimination preserves all integral coefficients and givesThis is the integral echelon basis of level-one modular forms.