How to see there are no nontrivial cusp forms for $\Gamma_0(4)$ of weight 2
I am searching for a proof of this that directly uses the provided set of generators for $M_2(\Gamma_0(4))$.
As mentioned in the other post, any cusp form in such space seems to be of the form $\lambda(3E_{2,2}-E_{2,4})$ for some $\lambda\in \mathbb{C}$.