Kohnen introduced the "plus" space as a subspace of the space of modular forms of half integral weight, first in his $\Gamma_0(4)$, Math. Annal.">1980 paper and then generalized the work in a later 1982 paper. Why is the condition $N$ odd, square-free and $\chi$ quadratic necessary?

To elaborate: Let $S_{k+1/2}\left(4N,\left(\frac{4\chi(-1)}{.}\right)\chi\right)$ denote the space of half-integral weight modular forms of level $4N$ and character $\left(\frac{4\chi(-1)}{.}\right)\chi$, where $\chi$ is a Dirichlet character of modulus $N$ and $\left(\frac{a}{b}\right)$ is the Kronecker symbol. Kohnen defines the plus sub-space $S_{k+1/2}^+\left(4N,\left(\frac{4\chi(-1)}{.}\right)\chi\right)$ by attaching certain conditions on the Fourier coefficients of the modular forms, and then develops nice theory analogous to the Atkin-Lehner-Li theory of newforms in the case of integral weight modular forms.

Question: Why are the conditions $N$ odd, sqaure-free and $\chi$ quadratic necessary in the second paper? Kohnen remarks that these are not necessary for few of the stated results, but I cannot figure out where exactly these are required.

The question is motivated by the fact that several authors (like Ueda, Yamana, Manickam-Ramakrishnan-Vasudevan) have generalized this work to other levels (like $8N,16N,32N$) and non-quadratic characters, and also to the full-space (a nice history can be found (but not restricted to) in the introduction of this paper and this paper); but this condition $N$ odd and square free is still there, and in some places $\chi$ is quadratic. I could not find the reason for these conditions.

Kohnen introduced the "plus" space as a subspace of the space of modular forms of half integral weight, first in his 1980 paper and then generalized the work in a later 1982 paper. Why is the condition $N$ odd, square-free and $\chi$ quadratic necessary?

To elaborate: Let $S_{k+1/2}\left(4N,\left(\frac{4\chi(-1)}{.}\right)\chi\right)$ denote the space of half-integral weight modular forms of level $4N$ and character $\left(\frac{4\chi(-1)}{.}\right)\chi$, where $\chi$ is a Dirichlet character of modulus $N$ and $\left(\frac{a}{b}\right)$ is the Kronecker symbol. Kohnen defines the plus sub-space $S_{k+1/2}^+\left(4N,\left(\frac{4\chi(-1)}{.}\right)\chi\right)$ by attaching certain conditions on the Fourier coefficients of the modular forms, and then develops nice theory analogous to the Atkin-Lehner-Li theory of newforms in the case of integral weight modular forms.

Question: Why are the conditions $N$ odd, sqaure-free and $\chi$ quadratic necessary in the second paper? Kohnen remarks that these are not necessary for few of the stated results, but I cannot figure out where exactly these are required.

The question is motivated by the fact that several authors (like Ueda, Yamana, Manickam-Ramakrishnan-Vasudevan) have generalized this work to other levels (like $8N,16N,32N$) and non-quadratic characters, and also to the full-space (a nice history can be found (but not restricted to) in the introduction of this paper and this paper); but this condition $N$ odd and square free is still there, and in some places $\chi$ is quadratic. I could not find the reason for these conditions.