Analogous theorem for Hilbert modular forms

Question

I have studied modular forms and saw a correspondence like a newform correspond to a automorphic representation of $\mathrm{GL}_n(\mathbb{A_Q})$. Does any similar result holds for Hilbert modular forms?

Also we know $S_k(\Gamma_1(M))= \bigoplus\nolimits_{\epsilon:(\mathbb{Z}/n\mathbb{Z})^*\to\mathbb{C}^*} S_k(\Gamma_0(M),\epsilon)$ and $\dim_{\mathbb{C}}S_k(\Gamma_1(M))$ ~ $M^2$ where $\Gamma_1(M)$ and $\Gamma_0(M)$ are well-known congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$. $S_k(\Gamma_1(M))$ is the subspace of holomorphic cusp forms and $S_k(\Gamma_0(M),\mathbb{\epsilon})$ is the subspace of holomorphic cusp forms having character $\epsilon$. I want to know again if there are similar results for Hilbert modular forms or Hilbert Cusp forms.

If yes, please suggest some good references for this.

Answer 1

There's a nice summary of the newform theory of Hilbert modular forms in Section 3 of the paper Twists of Hilbert Modular Forms by Thomas R. Shemanske and Lynne Walling.

As for your decomposition of the space of cusp forms, I mention a generalization of it to Hilbert modular forms towards the end of Section 2 of my paper Characterizing adelic Hilbert modular cusp forms by coefficient size. There's likely a much friendlier reference for these decompositions, but unfortunately it's been quite a while since I thought about this stuff and don't know of one off the top of my head.