I'm reading Iwaneic's "Topics in classical automorphic forms". Now, I'm reading the proof the theorem that for any Hecke character $\xi$ of a quadratic field $K/\mathbb{Q}$, there exists a $\mathrm{GL}_{2}/\mathbb{Q}$ automorphic form (holomorphic modular form or Maass form), which is a special case of automorphic induction. The proof uses Weil's converse theorem and the functional equation of the Hecke $L$-function. The author proved the case of modular forms, i.e. when $K$ is an imaginary quadratic field and left the real case as an exercise.
For the real case, let $\xi$ is a Hecke character of $K = \mathbb{Q}(\sqrt{D})$ where $D = \mathrm{disc}(K) >0$. Let $\mathfrak{m}$ be a modulus of $\xi$ and $\xi((a)) = (a/|a|)$ or $(a'/|a'|)$ ($a'$ is a conjugate of $a$) for $a\equiv 1$ (mod $\mathfrak{m}$). Then the theorem claims that $$ f(z) = \sum_{\mathfrak{a}\subseteq \mathcal{O}_{K}}\xi(\mathfrak{a})e^{2\pi i (N\mathfrak{a})z} $$ is a modular form of weight 1 on $\Gamma_{0}(|D|\cdot N\mathfrak{m})$. I tried to imitate the proof of the imaginary case, so I defined $$g(z) = C\sum_{\mathfrak{a}} \overline{\xi}(\mathfrak{a})e^{2\pi i (N\mathfrak{a})z}$$ for some appropriate constant $C$ and tried to prove that $g = f|_{\omega}$ where $\omega = \begin{pmatrix} & -1 \\ N & \end{pmatrix}$ and some other twisted functional equations. However, in the book, it only states the Weil's converse theorem for even positive weight $k$, so I'm not sure whether it also holds for $k=1$. (It seems that it works, according to this note.) Actually, I can't find where the author used the condition $k\geq 2$ in the proof of Weil's converse theorem.