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Start from the integral triangular basis of level-one modular forms. Proceed downwards in $j$. Set $g_{d-1}=F_{d-1}$; once $g_{j+1},\dots ,g_{d-1}$ are constructed, subtract from $F_j$ its coefficients at $q^{j+1},\dots ,q^{d-1}$ times these respective forms. The already constructed rows have a single nonzero coefficient among those initial positions, so this removes every unwanted initial coefficient without changing the coefficient one at $q^j$. All subtractions use integers. The resulting integral echelon basis of level-one modular forms therefore hasIts first $d$ coefficient vectors are the standard coordinate vectors, hence it is a basis. If another form had the same first $d$ coefficients as $g_j$, their difference would have those coefficients all zero; the triangular basis proves that difference vanishes. This proves uniqueness of every $g_j$, not merely uniqueness up to basis change.

For any $f\in M_k$, its expansion in this basis is $f={\sum }_{j=0}^{d-1}{a}_j(f)g_j$, since its first $d$ coefficients determine the basis coordinates. Comparing subsequent coefficients givesThe coefficients of $f$ need not be integral for this last identity. When needed at earlier indices, extend the notation by $c_n(j)={\delta }_{nj}$ for $0\le n<d$. This agrees with the actual coefficients of the basis forms and is the convention needed for the next part's all-index wording.

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.