Twisted modular forms of half-integral weight

Question

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 !

Answer 1

In general, the statement is something like the following. (The following is Proposition 3.12 from Ken Ono's book "The Web of Modularity")

Suppose that $g(z) = \sum c(n) q^{n}$ is a half-integer weight modular form for $\Gamma_{0}(4N)$ with character $\chi$. If $\psi$ is a Dirichlet character modulo $m$, then $$ \sum \psi(n) c(n) q^{n}$$ is a half-integer weight modular form for $\Gamma_{0}(4Nm^{2})$ with character $\chi \psi^{2}$.

Ono does not provide a proof of this. However, there is a proof given in Shimura's 1973 Annals paper "On modular forms of half-integral weight". The fact you are seeking is Lemma 3.6 on page 466.