A question related to Hilbert modular form

Results of this form are best stated adèlically. Perhaps the canonical reference is this paper of Shalika and Tanaka:

https://doi.org/10.2307/2373316

Sadly the paper was written pre-Jacquet-Langlands and is rather hard to read. Jacquet-Langlands do treat automorphic induction themselves in Section 12 of their seminal book:

http://doi.org/10.1007/BFb0058988

Alternatively, one can try reading this later paper of Labesse and Langlands, which discusses converses to automorphic induction:

https://doi.org/10.4153/CJM-1979-070-3

(See also my answer here: Reference for: CM Hilbert Modular forms arise from Hecke characters)

All of these deal with automorphic induction for Hecke characters; automorphic induction in more general settings is known due to the work of Arthur and Clozel:

https://www.jstor.org/stable/j.ctt1bd6kj6

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In what follows, I summarise the correspondence between Hecke characters and automorphically induced automorphic representations.

Let $E/F$ be a quadratic extension of number fields, and let $Omega$ be a unitary Hecke character of $mathbb{A}_E^{times}$, so that $Omega$ is the idèlic lift of a classical (primitive) Größencharakter $psi$ of $E$. This has a completed $L$-function $Lambda(s,Omega)$ whose finite part $L(s,Omega)$ has an Euler product of the form $$prod_{mathfrak{P}} frac{1}{1 - psi(mathfrak{P}) mathrm{N}_{E/mathbb{Q}}(mathfrak{P})^{-s}},$$ where the product is over the prime ideals $mathfrak{P}$ of $mathcal{O}_E$. Note that $psi(mathfrak{P}) = 0$ whenever $mathfrak{P}$ divides the conductor $mathfrak{Q}$ of $Omega$.

Automorphic induction associates to $Omega$ an automorphic representation $pi = pi(Omega)$ of $mathrm{GL}_2(mathbb{A}_F)$ whose completed $L$-function $Lambda(s,pi)$ is equal to $Lambda(s,Omega)$. (One can prove this via the converse theorem.)

Let $omega_{pi}$ denote the central character of $pi$, so that this is a Hecke character of $mathbb{A}_F^{times}$ that is the idèlic lift of a classical (primitive) Größencharakter $chi_{pi}$ of $F$; when $F = mathbb{Q}$, $chi_{pi}$ is just a Dirichlet character (it is the nebentypus of the newform associated to $pi$). One can check that $omega_{pi} = omega_{E/F} Omega|_{mathbb{A}_F^{times}}$, where $omega_{E/F}$ denotes the quadratic Hecke character associated to the quadratic extension $E/F$. Let $lambda_{pi}(mathfrak{n})$ denote the $mathfrak{n}$-th Hecke eigenvalue of $pi$, where $mathfrak{n}$ is an integral ideal of $mathcal{O}_F$. (Here I am normalising the Hecke eigenvalues as an analytic number theorist would, namely that $lambda_{pi}(mathfrak{p})$ is the sum of two complex numbers of absolute value $1$ when $mathfrak{p}$ does not divide the conductor of $pi$.) Then the finite part $L(s,pi)$ has an Euler product of the form $$prod_{mathfrak{p}} frac{1}{1 - lambda_{pi}(mathfrak{p}) mathrm{N}_{F/mathbb{Q}}(mathfrak{p})^{-s} + chi_{pi}(mathfrak{p}) mathrm{N}_{F/mathbb{Q}}(mathfrak{p})^{-2s}},$$ where the product is over prime ideals $mathfrak{p}$ of $mathcal{O}_F$. Note that the conductor $mathfrak{q}$ of $pi$ satisfies $mathfrak{q} = mathrm{N}_{E/F}(mathfrak{Q}) mathfrak{d}_{E/F}$, where $mathfrak{d}_{E/F}$ denotes the relative discriminant.

Now for each prime ideal $mathfrak{p}$, write $lambda_{pi}(mathfrak{p}) = alpha_{pi,1}(mathfrak{p}) + alpha_{pi,2}(mathfrak{p})$, where $alpha_{pi,1}(mathfrak{p}), alpha_{pi,2}(mathfrak{p})$ denote the Satake parameters. Note that $alpha_{pi,1}(mathfrak{p}) alpha_{pi,2}(mathfrak{p}) = chi_{pi}(mathfrak{p})$. Then by comparing Euler products, we have the following:

1. If $mathfrak{p}$ splits in $E$, so that $mathfrak{p} mathcal{O}_E = mathfrak{P} sigma(mathfrak{P})$ for some prime ideal $mathfrak{P}$ of $mathcal{O}_E$ with $mathrm{N}_{E/F}(mathfrak{P}) = mathrm{N}_{E/F}(sigma(mathfrak{P})) = mathfrak{p}$, where $sigma$ denotes the nontrivial Galois automorphism of $E/F$, then $alpha_{pi,1}(mathfrak{p}) = psi(mathfrak{P})$ and $alpha_{pi,2}(mathfrak{p}) = psi(sigma(mathfrak{P}))$.
2. If $mathfrak{p}$ is inert in $E$, so that $mathfrak{p} mathcal{O}_E = mathfrak{P}$ for some prime ideal $mathfrak{P}$ of $mathcal{O}_E$ with $mathrm{N}_{E/F}(mathfrak{P}) = mathfrak{p}^2$, then $alpha_{pi,1}(mathfrak{p}) = -alpha_{pi,2}(mathfrak{p}) = psi(mathfrak{P})^{1/2}$.
3. If $mathfrak{p}$ is ramified in $E$, so that $mathfrak{p} mid mathfrak{d}_{E/F}$ and $mathfrak{p} mathcal{O}_E = mathfrak{P}^2$ for some prime ideal $mathfrak{P}$ of $mathcal{O}_E$ with $mathrm{N}_{E/F}(mathfrak{P}) = mathfrak{p}$, then $alpha_{pi,1}(mathfrak{p}) = psi(mathfrak{P})$ and $alpha_{pi,2}(mathfrak{p}) = 0$.

From this and multiplicativity, one can deduce that $$lambda_{pi}(mathfrak{n}) = sum_{substack{mathfrak{N} subset mathcal{O}_E \ mathrm{N}_{E/F}(mathfrak{N}) = mathfrak{n}}} psi(mathfrak{N}).$$

I haven't yet described what happens at the archimedean places. At each archimedean place $w$ of $E$, the local component of $Omega$ is a unitary character $Omega_w : E_w^{times} to mathbb{C}^{times}$ with image in the unit circle.

1. If $E_w cong mathbb{R}$, then $Omega_w(x_w) = mathrm{sgn}(x_w)^{kappa_w} |x_w|_w^{it_w}$ for some $kappa_w in {0,1}$ and $t_w in mathbb{R}$. The local component of the completed $L$-function is $Gamma_{mathbb{R}}(s + kappa_w + it_w)$, where $Gamma_{mathbb{R}}(s) = pi^{-s/2} Gamma(s/2)$.
2. If $E_w cong mathbb{C}$, then $Omega_w(x_w) = e^{ikappa_w arg(x_w)} |x_w|_w^{it_w}$ for some $kappa_w in mathbb{Z}$ and $t_w in mathbb{R}$. The local component of the completed $L$-function is $Gamma_{mathbb{C}}(s + frac{|kappa_w|}{2} + it_w)$, where $Gamma_{mathbb{C}}(s) = 2(2pi)^{-s} Gamma(s)$.

From this, we can describe the local components of $pi$ at each archimedean place $v$ of $F$.

1. If $F_v cong mathbb{R}$ and $v$ splits in $E$ into two real places $w_1$ and $w_2$, then $pi_v$ is a principal series representation of the form $mathrm{sgn}^{kappa_{w_1}} |cdot|_v^{it_{w_1}} boxplus mathrm{sgn}^{kappa_{w_2}} |cdot|_v^{it_{w_2}}$.
2. If $F_v cong mathbb{R}$ and $v$ ramifies in $E$, so there exists a single complex place lying over $v$, then $pi_v$ is a discrete series representation of the form $D_{|kappa_w| + 1} otimes left|detright|_v^{it_w}$; in particular, the weight is $|kappa_w| + 1$.
3. If $F_v cong mathbb{C}$ then $v$ splits in $E$ into two complex places $w_1$ and $w_2$, and $pi_v$ is a principal series representation of the form $e^{ikappa_{w_1} arg} |cdot|_v^{it_{w_1}} boxplus e^{ikappa_{w_2} arg} |cdot|_v^{it_{w_2}}$.

Note that there are restrictions on the parameters $t_w$, since $Omega$ is trivial on $E^{times}$ and in particular on $mathcal{O}_E^{times}$.

(I write much of this down in section 4 of this paper of mine: https://doi.org/10.1093/imrn/rnx283)

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At this point, you know the Hecke eigenvalues of $pi$ and also all of its archimedean data. From here, you can write down explicitly the Fourier expansion of the newform of $pi$ (adèlically, this is its Whittaker expansion). Note that you need to be a little careful, since the constant term in the Fourier expansion does not necessarily vanish: $pi$ is cuspidal if and only if $Omega$ does not factor through the norm map; otherwise, the newform associated to $pi$ is an Eisenstein series.