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Using strong approximation (i.e. class number of $G$ respect to $\prod GL_2(\mathcal{O}_p)$ is one) one can prove that there is a unique cusp.

Suppose $G$ is the algebraic group defined by its rational points $G(\mathbb{Q})=GL_2(D)$ and $P$ is the standard $\mathbb{Q}$-parabolic subgroup of $G$. For an arithmetic subgroup $\Gamma\subset G(\mathbb{Q})$ the cosets $\Gamma\backslash G(\mathbb{Q})/P(\mathbb{Q})$ are called set of $\Gamma$-cusps.

Thanks. Regarding answer 1) I am little confused. What do you mean by $g.n(x)a(y)$? I mean what is the action of $g$ on $\begin{pmatrix}y&x\\&1\end{pmatrix}$ in matrix form?

Aren't Hecke operators always Hermitian with respect to Petersson inner product?

Thanks! I haven't realized it before. But the problem still is remaining as I noted that we can actually determine all $x_n(k)$'s. Kindly check the edited question.