Let A(\(\sigma\),\(\lambda\),\(\Gamma)\) denote the space of real-analytic automorphic eigenforms for a congruence subgroup \(\Gamma\) of the Hilbert modular group corresponding to a totally real algebraic number field F, with \(\sigma\) as (half-integral) weight and \(\lambda\) as eigenvalue. Let N(\(\sigma\),\(\lambda\),\(\Gamma)\) be the orthogonal complement in A(\(\sigma\),\(\lambda\),\(\Gamma)\) of the subspace of cusp forms. The principal result of the paper is that (for ''simple'' \(\lambda)\), the subspace N(\(\sigma\),\(\lambda\),\(\Gamma)\) can be generated, almost always, by Eisenstein series E(z,s;\(\rho)\) for suitably chosen parameters s, \(\rho\) ; in the exceptional cases, the derivatives \(\partial E/\partial s\) or the residues of E(z,s;\(\rho)\) need to be brought in as generators as well. In the case of holomorphic automorphic forms (which come under \(\lambda =0)\), these results subsume earlier work of Hecke, Kloosterman, Petersson, Shimizu and Pei; in the real-analytic case, the corresponding result for \(\sigma =0\), \(\lambda\geq 1/4\) and \(F={\mathbb{Q}}\) is due to Maass (see also subsequent work of Roelcke on eigenforms of arbitrary weight for Fuchsian groups). Actually, the author's proof uses a generalization of Maass' bilinear relations between coefficients of constant terms for eigenforms; also an essential role in the proof is played by the author's important recent work on Eisenstein series.