Fix positive integers $k, N$ and let $\omega$ be a Dirichlet character mod $N$.

Let $f\in S_k(N,\phi)$ be a normalized newform (i.e. of weight $k$, level $N$ and character $\phi$) with fourier expansion $\sum_{n\geq 1} a(n)q^n$. In her paper 'Newforms and Functional Equations', Winnie Li showed that if $q$ is a prime dividing $N$ and $\phi$ is not a character mod $N/q$, then $|a(q)|=q^{\frac{k-1}{2}}$. In particular, $a(q)\neq 0$. This is Theorem 3.(ii) of the paper. I should also note that special cases of this were proven earlier Atkin-Lehner, Hecke and Ogg.

I am interested in the Hilbert modular version of this result. More specifically, let $\mathfrak{N}$ be an integral ideal of a totally real number field $K$ of degree $d$ over $\mathbb{Q}$, $\Phi$ be a Hecke character induced by a numerical character mod $\mathfrak{N}$ and $k\in (\mathbb{Z}_+)^d$.

Let $S_k(\mathfrak{N},\Phi)$ be the space of Hilbert modular cusp forms of weight $k$, level $\mathfrak{N}$ and character $\Phi$, viewed adelically to circumvent class number issues (I make no assumptions regarding the class number of $K$). Shimura, in 'The special values of the zeta functions associated to Hilbert modular forms' defined "Fourier coefficients" $C(\mathfrak{m},\textbf{f})$ for cusp forms $\textbf{f}\in S_k(\mathfrak{N},\Phi)$ (this on the bottom of page 649 of the paper).

Let $\textbf{f}\in S_k(\mathfrak{N},\Phi)$ be a normalized newform. I am interested in knowing when $C(\mathfrak{q},\textbf{f})=0$, where $\mathfrak{q}\mid\mathfrak{N}$. In their paper 'Twists of Hilbert Modular Forms', Tom Shemanske and Lynne Walling showed that if the numerical character inducing $\Phi$ is not defined mod $\mathfrak{N}/ \mathfrak{q}$, then this coefficient is nonzero whenever $\mathfrak{q}$ has degree 1 over the rationals or $\mathfrak{q}$ exactly divides $\mathfrak{N}$ (this is part 2 of Theorem 3.3 in the paper). In a remark directly after the Theorem, Tom and Lynne say that the reason for the restrictions on the prime $\mathfrak{q}$ is that this is what is needed to make Ogg's proof in the classical case go through. The 'truth', however, is that the $\mathfrak{q}$-th coefficient should always be nonzero in this case, without any restrictions on the degree of the prime $\mathfrak{q}$. A referee told Tom (who is my advisor) that the representation theory implies this stronger result. Tom lost the referee's report some time ago however, and doesn't know of any reference for this representation theoretic result. Does anyone know of a paper in which this result is proven without any restrictions on the prime $\mathfrak{q}$?

$\textbf{Question}:$ Let $\textbf{f}\in S_k(\mathfrak{N},\Phi)$ be a normalized newform, $\mathfrak{q}$ be a prime dividing $\mathfrak{N}$ and suppose that the numerical character inducing $\Phi$ is not defined mod $\mathfrak{N}\mathfrak{q}^{-1}$. Is it true that $C(\mathfrak{q},\textbf{f})\neq 0$?