Let \(f=\sum_{n\geq 1} a(n)q^n\) be a cusp form of weight \(k\) and character \(\chi\) for the group \(\Gamma_0(n)\), where \(N\geq 1\) is an integer. Let \(K\) be a real quadratic field, \(D\) the discriminant of \(K\), \(\mathcal O_K\) the ring of integers of \(K\) and \(\mathfrak d=\frac 1{\sqrt D} \mathcal O_K\) the different. Set \[ E_f^K(z_1,z_2)=\sum_{{\nu\in\mathfrak d^{-1}}\atop {\nu\gg 0}} e^{2i\pi(\nu z_1+\nu' z_2)} \sum_{{d\mid \bigl(\nu\mathfrak d\bigr)}\atop {d\in\mathbb N^*}} d^{k-1}\chi(d)\left({4D\over d}\right) a\left(\eta_{K/\mathbb Q}\left({\nu\mathfrak d\over d}\right)\right). \]
Then if \(k\geq 3\) is an integer, we show that \(E_f^K\) is a Hilbert modular form of weight \(k\) and character \(\chi\cdot\eta_{K/\mathbb Q}\) on a congruence subgroup of the Hilbert modular group. The proof depends on G. Shimura’s theorem on forms of half-integral weight [Ann. Math. (2) 97, 440–481 (1973; Zbl 0266.10022)] and on a combinatorial theorem of the author [Math. Ann. 217, 271–285 (1975; Zbl 0311.10030)]. Furthermore, using a deep theorem of Vaserštein, we describe precisely the congruence subgroup occurring above.