I am looking for references (or explainations) about the twist of modular forms of half-integral weight. I try to mimic the proof of the "integral weight case" to prove that the twist of $$ \theta(\tau)=\sum_{n\in \textbf{Z}}{q^{n^2}} \quad (\tau \in \mathcal{H}) $$ by a Dirichlet character $\chi$ of odd modulus, namely $$ \theta_{\chi}(\tau)=\sum_{n\in \textbf{Z}}{\chi(n^2)q^{n^2}} \quad (\tau \in \mathcal{H}), $$ is still a modular form. But this method highly depends on the form of the automorphic factor of the $\theta$ function.

1. Do we know if the twist of a modular form of half-integral weight (defined by the use of a four-sheeted cover of $\operatorname{Gl}_2^{+}(\textbf{Q})$) is still a modular form ?

2. If $\eta$ denotes the usual Dedekind function, is $\eta_{\chi}$ a modular form ?

Many thanks !