Let $k \geq 2$. Say $S_{k}^{\text{new}}(\Gamma_{0}(N), \chi)$ is one-dimensional and spanned by a newform with CM, and $S_{k}^{\text{old}}(\Gamma_{0}(N), \chi)$ has positive dimension. Must it be true that $S_{k}^{\text{new}}(\Gamma_{0}(N), \chi)$ is spanned by a Poincare series?
Obviously this is true if $S_{k}(\Gamma_{0}(N), \chi)$ is itself one-dimensional. The Poincare series span the whole cuspidal space, and so there is a basis of Poincare series.
Edit: This isn't necessarily true when the unique $g \in S_{k}^{\text{new}}(\Gamma_{0}(N), \chi)$ doesn't have CM. I have added this detail.
The point of $$P_{\Gamma_0(N),\chi,k,n}(z)=\sum_{\gamma \in \langle T\rangle \backslash \Gamma_0(N)} \chi(\gamma)\gamma'(z)^{k/2} \exp(2i\pi n \gamma(z))$$ is that if $f\in M_k(\Gamma_0(N),\chi)$ then the Petersson inner product gives
$$\langle f,P_{\Gamma_0(N),\chi,k,n} \rangle = a_n(f)C_{N,k}$$ So $P_{\Gamma_0(N),\chi,k,1}$ is never orthogonal to $S_k(\Gamma_0(N),\chi)^{old}$ (not a newform), and the span of the $P_{\Gamma_0(N),\chi,k,n},n\ge 1$ represents all the linear maps $S_k(\Gamma_0(N),\chi)\to \Bbb{C}$ so $span(P_{\Gamma_0(N),\chi,k,n},n\ge 1)$ is the whole of $S_k(\Gamma_0(N),\chi)$.