For half integral weight newform does the Shimura correspondance give a newform?

Question

in number theory,
half integral wieght newforms implies the shimura correspondance is it also newforms ?
namely,
if $f(z)=\sum_{n=1}^{\infty}a(n)q^n\in S_{k+1/2}^{new}(N,\chi)$ is half integral newforms then $f_t(z)=\sum_{n=1}^{\infty}a_t(n)q^n\in S_{2k}(\Gamma(2N),\chi^2)$ given by the shimura correspendance with \begin{equation} a_{t}(n)=\underset{d\mid n}{\sum}\chi_{t,N}(d)d^{k-1}a(\frac{n^2t}{d^2})\label{eq1} \end{equation} where $\chi_{t,N_i}$ denotes the character defined by $\chi_{t,N}(d):=\chi(d)(\frac{(-1)^{k}N^2t}{d})$. is it $f_t$ also newforms ?