Let $M_k(N,\chi)$ be the space of modular forms with integer weight $k$, level $N$ and Dirichlet character $\chi$ modulo $N$. Further assume that $\chi$ is primitive. Why there are no oldforms in this space?
Attempt: Let $M|N$ and $d|(\frac{N}{M}).$ If $f\in M_k(M),$ then $f(dz)\in M_K(N),$ and $f$ is an 'old' form for the space $M_k(N)$. In a similar fashion when a Drichlet character $\chi$ is introduced, if $f\in M_k(N,\chi)$, then it cannot come from a 'lower' space $M_k(M,\psi)$ - because $\chi$ is primitive and it cannot be decomposed to get a 'lower' $\psi$. I know I am being vague here, still is this correct or missing something?
In a similar way, does the space $M_{k+1/2}(N,\chi)$ consists only of newforms when $\chi$ is primitive (modulo $N$)?