In this paper, we compute for odd fundamental discriminants D>1 the Fourier
expansion of non-holomorphic elliptic Eisenstein series for Ξ0β(D) with
quadratic nebentypus character ΟDβ satisfying a certain plus space
condition. For each genus of Q(Dβ), 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.Comment: 12 page