Let $$\phi(z)=\sum_{\gamma\in U_2(\mathbb{Z})\backslash SL(2,\mathbb{Z})} \,\,\,\,\,\sum_{m_1=1}^{\infty}\,\,\sum_{m_2\neq 0}a(m,n)W_{\text{Jacquet}}\left(\begin{pmatrix} |m_1m_2| & & \\ & m_1 & \\ & & 1 \end{pmatrix}\begin{pmatrix}\gamma & \\ & 1\end{pmatrix}z\,,\, \nu,\,\psi_{1,\frac{m_2}{|m_2|}} \right)$$ be a Maass cusp form of type $\nu=(\nu_1,\nu_2)$ for $SL(3,\mathbb{Z})$. Show that $a(m,n)=O(1).$
Note that$$a(m,n)W_{\text{Jacquet}}\left(\begin{pmatrix} |m_1m_2|y_1y_2 & & \\ & m_1y_1 & \\ & & 1 \end{pmatrix}\,,\, \nu,\,\psi_{1,\frac{m_2}{|m_2|}} \right)\\=\int_0^1\int_0^1\int_0^1\phi(z)e^{-2\pi i[m_1x_1+m+2x_2]}\,dx_1dx_2dx_3$$ If we knew that $\phi$ is bounded, then the claim would follow by choosing $y_1=c_1|m_1|^{-1}, y_2=c_2|m_2|^{-1}$ , but we only know that $\phi(z)$ is square integrable in the Seigel domain.