Let $a=(a_1,a_2)\in \mathbb{Z}/4\mathbb{Z}$. For $a\neq(0,0)$ and $k>2$ define the Eisenstein series for $\Gamma(4)$ by $$G_k^a(z)=\sum_{n\equiv a(\text{mod 2})}(n_1z+n_2)^{-k}$$
Show that $$F_k^a(z)=\sum_{m=0}^3G_k^{(a_1,a_2+ma_1)}(z)$$ is a modular form for $\Gamma_1(4)$ , and evaluate $F_k^a(z)$ at each cusps of $\Gamma(4)$ (I know the cusps of $\Gamma(4)$ are $0,-1/2,\infty$).