Oldform by argument dilation ($f(z)⟼f(Dz)$)

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.

The group ${\Gamma }_0(3)$ has index four. The dimension bound for modular forms on a finite-index subgroup gives dimension at most two in weight four. The Eisenstein series $E_4(z)$ and its oldform by argument dilation $E_4(3z)$ are independent, since their constant coefficients agree while their $q$ coefficients are $240$ and zero.