This is referring to Example 9.15 in William Stein's book 'Modular forms: a computational approach'.
In this example, we are to calculate the newform of weight 2 level 23 in $S_{2}(\Gamma_{0}(23))$. It starts by working out the Manin symbols $(0,0),(1,0),(0,1)$ and the matrix form of Hecke operator $T_{2}$. We find that $T_{2}$ has an eigenspace (ker($T_{2}-3$)) spanned by the Eisenstein series of level 23 weight 2, and the other ($V =$ker($T_{2}^{2}+T_{2}-1$)) corresponds to $S_{2}(\Gamma_{0}(23))$.
Then the algorithm goes on to projecting onto $V$, which get me confused when he says the following matrix $$ \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & -2/11\\ 0 & 1 & -3/11 \end{pmatrix} $$ has the first two columns being the 'echelon basis for $V$' and last column being 'echelon basis for the Eisenstein subspace'. I have looked through the related sections in the book and I am still confused about where this comes from, and how we may calculate such a matrix for general levels (say $N = 11,37,etc$).
