A Jacobi-Eisenstein series of weight two on congruence Jacobi subgroup (Q625853)

For \(k,m\) positive integers let \[ E_{k,m;4}(\tau,z):=\frac12 \sum_{\substack{ c,d\in{\mathbb Z}\\ 4|c, (c,d)=1}} \sum_{\lambda\in{\mathbb Z}}(c\tau+d)^{-k}e^{2\pi i m\left(\lambda^2\frac{a\tau+b}{c\tau +d}+2\lambda\frac{z}{c\tau +d}-\frac{cz^2}{c\tau +d}\right)}. \] The authors compute explicitly the Fourier coefficients of \(E_{2,1;4}\).