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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.

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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.

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    $\begingroup$ Another good reference for seeing how Hilbert modular forms relate to automorphic forms/representations is Gelbart's "Automorphic forms on adele groups". $\endgroup$ – Leray Jenkins Jan 20 at 9:19
  • $\begingroup$ @LerayJenkins Thanks though it does not talk much about the relation between classical and adelic Hilbert modular forms. $\endgroup$ – Kiddo Jan 21 at 14:52

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