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I am making a latex beamer presentation and I have a table. My table contains equation in math modes and looks like

enter image description here

In the last part, Factorization, I want to align the second Delta of chi twelve with the one from psi twelve from the last line. When this is done, I want to align the arrow from chi ten with the arrow from chi10. I tried many thing but nothing worked out right now. Here is my code

\documentclass{beamer}
\usepackage[french]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{multirow}
\usepackage[font=small,skip=0pt]{caption}
\usepackage{tabu}
\usetheme{Warsaw}
\usecolortheme{seahorse}
\begin{document}

\begin{frame}
\frametitle{Siegel modular forms}
\begin{center}



  \tabulinesep=1.2mm
 \begin{tabu}{|c| c| c|}
 \hline %inserts horizontal line
 &Genus 1 &  Genus 2 \\[0.5ex] 
 \hline 
 Generators  & $\phi_4$, $\phi_6$ & $\psi_4,\psi_6,\chi_{10},\chi_{12}$ \\
\hline
 Siegel operator & $\Phi(\phi_4) = \Phi(\phi_6) = 1 $& $\Phi(\psi_i)=\phi_i,      \Phi(\chi_{j})= 0$\\
 \hline
 Discriminant & $\Delta =\frac{\phi_4^3-\phi_6^2}{1728} = \eta^{24}$ &   $\psi_{12} = \frac{\psi_4^3-\psi_6^2}{1728}$ 
  \\\hline
  Siegel operator & $\Phi(\Delta)=0$ $\rightarrow$ cusp form &   $\Phi(\psi_{12})=\Delta$\\ \hline  
\multicolumn{3}{c}{}\\[-0.7em]
\hline
Factorization & \multicolumn{2}{c|}{$\begin{array} {lcllcl} \psi_4 &    \rightarrow &\phi_4\otimes\phi_4 &\chi_{10}&\rightarrow &0\\ \psi_6 &   \rightarrow & \phi_6\otimes\phi_6 & \chi_{12} & \rightarrow &   \Delta\otimes\Delta\\
 \psi_{12} & \rightarrow &  \multicolumn{4}{l}{\phi_4^3\otimes \Delta +  \Delta \otimes \phi_4^3-1728\Delta\otimes\Delta }
\end{array}$}\\ 
\hline  
\end{tabu}
\end{center}
\end{frame}

\end{document}

Thank you very much.

4

enter image description here

\documentclass{article}
\begin{document}
\(\begin{array}{l@{\;}lr@{}l}
    \psi_4    & \rightarrow \phi_4\otimes\phi_4 & \chi_{10}  \rightarrow {} & 0\\
    \psi_6    & \rightarrow \phi_6\otimes\phi_6 & \chi_{12}  \rightarrow {} & \Delta\otimes\Delta\\
    \psi_{12} & \multicolumn{2}{@{}l@{}}{\rightarrow \phi_4^3\otimes \Delta +
                                              \Delta \otimes \phi_4^3-1728} & \Delta\otimes\Delta
  \end{array}
\)
\end{document}
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