Elliptic Eisenstein series associated to ideals in real quadratic number fields

In this paper, we compute for odd fundamental discriminants $D>1$ the Fourier expansion of non-holomorphic elliptic Eisenstein series for ${\mathrm{\Gamma }}_0(D)$ with quadratic nebentypus character ${\chi }_D$ satisfying a certain plus space condition. For each genus of $\mathbb{Q}(\sqrt{D})$, we obtain an associated plus space condition and corresponding Eisenstein series in all positive even weights. In weight $k=2$, the Fourier coefficients are associated to the geometry of Hirzebruch–Zagier divisors on Hilbert modular surfaces.