This is a question related to Hilbert Modular Form.

Let $\mathbb{K}=\mathbb{Q}(\sqrt D)$ be an imaginary quadratic with discriminant $D<0$ and $\zeta (\text{mod } m)$ a Hecke character such that $$\zeta((a))= \left( \frac{a}{|a|} \right)^u \text{ if } a \equiv 1 (\text{mod }m)$$ where $u$ is any non-negative integer.Then, $$ f(z)= \sum_a \zeta(a) N_{\mathbb{K}/\mathbb{Q}}(a)^\frac{u}{2} e(zN_{\mathbb{K}/\mathbb{Q}}(a))\in M_k(\Gamma_0(N),\chi) $$ where $k=u+1, N= |D|N_{\mathbb{K}/\mathbb{Q}}(m)$ and $\chi (\text{mod } m)$ is the Dirichlet character given by $$ \chi(n)= \chi_D(n) \text{ if } n\in \mathbb{Z} .$$ Moreover $f$ is a cusp form if $u>0$. This is a theorem from the book $``\text{ Topics in classical automorphic form}" $ by Henryk Iwaniec(page 213). This theorem actually provides a connection between conductor of a Hecke character and level and weight of a modular form by Automorphic induction.

I would like to know if there is a similar thereom in the case of Hilbert modular forms too that is if a theorem precisely shows how to construct a hilbert modular form over a totally real field $\mathbb{F}$ from a Hecke character of an imaginary quadratic extension $\mathbb{E}$ of $\mathbb{F}$.

Thank you in advance.

This is a question related to Hilbert modular forms.

Let $\mathbb{K}=\mathbb{Q}(\sqrt D)$ be an imaginary quadratic field with discriminant $D<0$ and $\zeta (\text{mod } m)$ a Hecke character such that $$\zeta((a))= \left( \frac{a}{|a|} \right)^u \text{ if } a \equiv 1 \pmod{m}$$ where $u$ is any non-negative integer. Then $$ f(z)= \sum_a \zeta(a) N_{\mathbb{K}/\mathbb{Q}}(a)^\frac{u}{2} e(zN_{\mathbb{K}/\mathbb{Q}}(a))\in M_k(\Gamma_0(N),\chi) $$ where $k=u+1, N= |D|N_{\mathbb{K}/\mathbb{Q}}(m)$ and $\chi (\text{mod } m)$ is the Dirichlet character given by $$ \chi(n)= \chi_D(n) \text{ if } n\in \mathbb{Z} .$$ Moreover $f$ is a cusp form if $u>0$. This is a theorem from the book "Topics in Classical Automorphic Forms" by Henryk Iwaniec  (page 213). This theorem actually provides a connection between the conductor of a Hecke character and the level and weight of a modular form by automorphic induction.

I would like to know if there is a similar thereom in the case of Hilbert modular forms too; that is, if a theorem precisely shows how to construct a hilbert modular form over a totally real field $\mathbb{F}$ from a Hecke character of an imaginary quadratic extension $\mathbb{E}$ of $\mathbb{F}$.

Thank you in advance.