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Let $D$ be a definite quaternion division algebra over $\mathbb{Q}$ and $\mathcal{O}$ be an Eichler order of $D$. Let $F$ be a Maass form in $L^2(PGL_2(\mathcal{O})\backslash PGL_2(D\otimes\mathbb{R}))$. For a finite unramified prime $p$ we have $K_p=GL_2(\mathcal{O}_p)=GL_4(\mathbb{Z}_p)$. We can think of a Hecke operator of $GL_2(D_p)$ with respect to $GL_2(\mathcal{O}_p)$ as: $$p^2T_{p,2}=K_p\begin{pmatrix}p&&&\\&p&&\\&&1&\\&&&1\end{pmatrix}K_p,$$which is self-adjoint. Let, $T_{p,2}F=\lambda(p)F$, so $\lambda(p)$ is real.

One can calculate the Fourier expansion of $T_{p,2}F$. Namely if $\rho(n)$ are Forier coefficient of $F$ then, $$\lambda(p)\rho(n)=\rho(p^{-1}n)+\rho(pn)+\frac1p\sum_{\alpha,\beta\in C_p}\rho(\alpha^{-1}n\beta),$$ where $C_p=\{\alpha|\text{reduced norm}(\alpha)=p\}/\mathcal{O}^\times$ (see prop 5.9, 2c). Hence one can calculate,$$\lambda(p)\rho(1)=\rho(p)+(p+\frac1p)\rho(1).$$

$\textbf{Q.}$ Is it true that $\lambda(p)\gg1$ for almost all finite $p$? I.e. Can we say something about order of $\rho(p)$?

I am not sure whether involving Jacquet-Langlands makes this question easier to deal with or not.

Thanks for any help and/or reference.

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