Weight-four Eisenstein basis at level two ($M_4({\Gamma }_0(2))=\mathbb{C}{E}_4(\tau )\oplus \mathbb{C}{E}_4(2\tau )$)

There are no weight-four cusp forms at level two. A nonzero form would have a weight-twelve coset norm of a modular form with modular cusp order at least two, contradicting the simple modular cusp order and interior nonvanishing of the modular discriminant. The two independent displayed Eisenstein series span the weight-four space, since its cusp-constant map embeds it in a two-dimensional space.