Nonfreeness of some algebras of hermitian modular forms

We study the algebras of hermitian automorphic forms for the lattice $L_n=diag(1,1,\dots ,1,-1)$ and for the field $K=\mathbb{Q}(\sqrt{-d})$ such that $p=2$ is unramified and the ring of integers ${\mathcal{O}}_K$ is a p.i.d. We prove that for $d>7$ these algebras can’t be free. When $d=7$ and $d=3$ we give an estimate for the dimension of the symmetric spaces for which these algebras might be free. We also compare our results with the known results for $d=3$.