Weight-four modular forms on Gamma 0 3 ($M_4({\Gamma }_0(3))=\mathbb{C}{E}_4(z)\oplus \mathbb{C}{E}_4(3z)$)

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.