Oldform by argument dilation
= Oldform by argument dilation
{title2=$f(z)\longmapsto f(Dz)$}
If $N_1D\mid N$ and $f\in M_k(\Gamma_0(N_1))$, then $f(Dz)\in M_k(\Gamma_0(N))$. Conjugating by $\operatorname{diag}(D,1)$ 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.