Free algebras of the Hilbert modular forms

Let $d>0$ be square-free integer and $L_d$ be the Hilbert lattice, i.e. the even lattice of signature (2, 2) such that $L_d=\begin{pmatrix}0 & 1 \\1 & 0\\ \end{pmatrix} \oplus \begin{pmatrix} 2 & 1\\1 & \frac{1-d}{2}\\ \end{pmatrix}$ when $d=1 \pmod{4}$, or $L_d = \begin{pmatrix}0 & 1 \\1 & 0 \\ \end{pmatrix} \oplus \begin{pmatrix} 2 & 0 \\0 & -2d \\ \end{pmatrix}$ when $d=2,3\pmod{4}$. Consider $\Gamma_d=O^+(L_d)$ and denote by $A(\Gamma_d)$ the algebra of $\Gamma_d$-automorphic forms. The main goal of the report is the following Theorem: If the algebra $A(\Gamma_d)$ is free then $d \in \{2,3,5,6,13,21\}$.