 Oldform by argument dilation

If $N_{1}D\mid N$ and $f\in M_k({\Gamma }_0(N_1))$, then $f(Dz)\in M_k({\Gamma }_0(N))$. Conjugating by $\mathrm{diag}(D,1)$ changes a lower-left matrix entry $c$ to $c/D$, which is divisible by $N_1$. A rational upper-triangular factorization at every cusp preserves boundedness and hence cusp holomorphy.