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\begin{document}
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\begin{tabular}{c|c|c|c} 
 $i$    &   $f_i(\Omega)$             & $\Gamma^{L, {\rm tr}}_i( q^2\xspace)/{\bf k} q^2\xspace $ & $\eta^{L\rightarrow\xspace R}_i$  \\ \hline \hline
 1   &   $P^0_0 Y^0_0$     &  $\left[  |H^L_0|^2\xspace +  |H^L_\parallel|^2\xspace +  |H^L_\perp|^2\xspace +  |S^L|^2\xspace +  |D^L_0|^2\xspace +  |D^L_\parallel|^2\xspace +  |D^L_\perp|^2\xspace\right]$ & $+1$ \\ \hline 
 2   &   $P^0_1 Y^0_0$     &  $2\left[\frac{2}{\sqrt{5}}  Re(H^L_0 D^{L\ast}_0)\xspace +  Re(S^L H_0^{L \ast})\xspace + \sqrt{\frac{3}{5}}  \, Re( H^L_\parallel D^{L\ast}_\parallel + H^L_\perp D^{L\ast}_\perp  )\right]$ & $+1$ \\ \hline 
 3   &   $P^0_2 Y^0_0$     &  $\frac{\sqrt{5}}{7}$ ( $ |D^L_\parallel|^2$ \xspace +  $ |D^L_\perp|^2$ \xspace) - $\frac{1}{\sqrt{5}}$ ( $ |H^L_\parallel|^2$ \xspace +  $ |H^L_\perp|^2$ \xspace) + $\frac{2}{\sqrt{5}}$  $ |H^L_0|^2$ \xspace  + $\frac{10}{7\sqrt{5}}$  $ |D^L_0|^2$ \xspace + $2$  $ Re(S^L D_0^{L \ast})$ \xspace & $+1$ \\  \hline
 4   &   $P^0_3 Y^0_0$     &  $\frac{6}{\sqrt{35}} \left[ - \, Re(H^L_\parallel D^{L\ast}_\parallel +  H^L_\perp D^{L\ast}_\perp)  + \sqrt{3}  Re(H^L_0 D^{L\ast}_0)\xspace  \right]$ & $+1$\\  \hline
 5   &   $P^0_4 Y^0_0$     &  $\frac{2}{7} \left[ -2 ( |D^L_\parallel|^2\xspace +  |D^L_\perp|^2\xspace) + 3  |D^L_0|^2\xspace \right] $ & $+1$\\  \hline
 6   &   $P^0_0 Y^0_2$     &  $\frac{1}{2 \sqrt{5}} \left[ ( |D^L_\parallel|^2\xspace +  |D^L_\perp|^2\xspace) + ( |H^L_\parallel|^2\xspace +  |H^L_\perp|^2\xspace) - 2  |S^L|^2\xspace - 2  |D^L_0|^2\xspace - 2  |H^L_0|^2\xspace \right]$ & $+1$ \\  \hline
 7   &   $P^0_1 Y^0_2$     &  $\left[ \frac{\sqrt{3}}{5} \, Re(H^L_\parallel D^{L\ast}_\parallel  + H^L_\perp D^{L\ast}_\perp) - \frac{2}{\sqrt{5}} \, Re(S^L H^{L\ast}_0)  - \frac{4}{5} \, Re(H^L_0 D^{L\ast}_0)\right] $ & $+1$  \\ \hline
 8   &   $P^0_2 Y^0_2$     &  $ \left[ \frac{1}{14} ( |D^L_\parallel|^2\xspace +  |D^L_\perp|^2\xspace) - \frac{2}{7}  |D^L_0|^2\xspace - \frac{1}{10} ( |H^L_\parallel|^2\xspace +  |H^L_\perp|^2\xspace) - \frac{2}{5}  |H^L_0|^2\xspace - \frac{2}{\sqrt{5}}  Re(S^L D_0^{L \ast})\xspace \right]$ & $+1$  \\  \hline
 9   &   $P^0_3 Y^0_2$     &  $ - \frac{3}{5 \sqrt{7}} \left[ \, Re( H^L_\parallel D^{L \ast}_\parallel + H^L_\perp D^{L \ast}_\perp) + 2 \sqrt{3} \, Re(H^L_0 D^{L \ast}_0 ) \right] $ & $+1$\\  \hline
 10  &   $P^0_4 Y^0_2$     &  $ -\frac{2}{7 \sqrt{5}}  \left[  |D^L_\parallel|^2\xspace +  |D^L_\perp|^2\xspace + 3  |D^L_0|^2\xspace \right] $ & $+1$  \\  \hline
 11  &   $P^1_1 \sqrt{2}\, Re(Y^1_2)$ &  $-\frac{3}{\sqrt{10}} \left[ \sqrt{\frac{2}{3}} \, Re(H^L_\parallel S^{L \ast}) - \sqrt{\frac{2}{15}} \, Re(H^L_\parallel D^{L \ast}_0  ) + \sqrt{\frac{2}{5}} \, Re(D^L_\parallel H^{L \ast}_0 ) \right] $  & $+1$\\  \hline
 12  &   $P^1_2 \sqrt{2}\, Re(Y^1_2)$ &  $-\frac{3}{5} \left[ \, Re( H^L_\parallel H^{L \ast}_0)  + \sqrt{\frac{5}{3}} \, Re (D^L_\parallel S^{L \ast})  + \frac{5}{7 \sqrt{3}} \, Re(D^L_\parallel D^{L\ast}_0) \right] $ & $+1$ \\  \hline
 13  &   $P^1_3 \sqrt{2}\, Re(Y^1_2)$ &  $-\frac{6}{5 \sqrt{14}} \left[2 \, Re(D^L_\parallel H^{L\ast}_0)  + \sqrt{3} \, Re(H^L_\parallel D^{L\ast}_0 ) \right] $ & $+1$ \\  \hline
 14  &   $P^1_4 \sqrt{2}\, Re(Y^1_2)$ &  $- \frac{6}{7\sqrt{2}} \, Re(D^L_\parallel D^{L\ast}_0)$  & $+1$ \\  \hline
 15  &   $P^1_1 \sqrt{2}\, Im(Y^1_2)$ &  $3 \left[ \frac{1}{\sqrt{15}} \, Im(H^L_\perp S^{L\ast}) + \frac{1}{5} \, Im(D^L_\perp H^{L\ast}_0) - \frac{1}{5 \sqrt{3}}  \, Im(H^L_\perp D^{L\ast}_0) \right]  $  & $+1$ \\  \hline
 16  &   $P^1_2 \sqrt{2}\, Im(Y^1_2)$ &  $ 3\left[ \frac{1}{7 \sqrt{3}} \, Im(D^L_\perp D^{L\ast}_0)  + \frac{1}{5} \, Im(H^L_\perp H^{L\ast}_0)  + \frac{1}{\sqrt{15}} \, Im(D^L_\perp S^{L\ast})   \right] $  & $+1$ \\  \hline
 17  &   $P^1_3 \sqrt{2}\, Im(Y^1_2)$ &  $\frac{6}{5 \sqrt{14}} \left[ 2 \, Im(D^L_\perp H^{L\ast}_0)  + \sqrt{3} \, Im(H^L_\perp D^{L\ast}_0) \right]  $   & $+1$ \\  \hline
 18  &   $P^1_4 \sqrt{2}\, Im(Y^1_2)$ &  $\frac{6}{7\sqrt{2}} \, Im(D^L_\perp D^{L\ast}_0)$  & $+1$ \\  \hline
 19  &   $P^0_0 \sqrt{2}\, Re(Y^2_2)$ &  $-\frac{3}{2\sqrt{15}} \left[ ( |H^L_\parallel|^2\xspace -  |H^L_\perp|^2\xspace) + ( |D^L_\parallel|^2\xspace -  |D^L_\perp|^2\xspace) \right] $  & $+1$ \\  \hline
 20  &   $P^0_1 \sqrt{2}\, Re(Y^2_2)$ &  $-\frac{3}{5} \left[ \, Re(H^L_\parallel D^{L\ast}_\parallel)   - \, Re(D^L_\perp H^{L\ast}_\perp) \right] $  & $+1$ \\  \hline
 21  &   $P^0_2 \sqrt{2}\, Re(Y^2_2)$ &  $\frac{\sqrt{3}}{2} \left[ - \frac{1}{7} ( |D^L_\parallel|^2\xspace -  |D^L_\perp|^2\xspace)   + \frac{1}{5} (  |H^L_\parallel|^2\xspace -  |H^L_\perp|^2\xspace ) \right] $  & $+1$ \\  \hline
 22  &   $P^0_3 \sqrt{2}\, Re(Y^2_2)$ &  $\frac{3}{5} \sqrt{ \frac{3}{7}} \left[ \, Re(H^L_\parallel D^{L\ast}_\parallel)   - \, Re(D^L_\perp H^{L\ast}_\perp) \right] $  & $+1$ \\  \hline
 23  &   $P^0_4 \sqrt{2}\, Re(Y^2_2)$ &  $\frac{2}{7} \sqrt{ \frac{3}{5}}  ( |D^L_\parallel|^2\xspace -  |D^L_\perp|^2\xspace) $ & $+1$ \\  \hline
 24  &   $P^0_0 \sqrt{2}\, Im(Y^2_2)$ &  $\sqrt{\frac{3}{5}} \left[ \, Im(H^L_\perp H^{L\ast}_\parallel) + \, Im(D^L_\perp D^{L\ast}_\parallel) \right] $   & $+1$ \\  \hline
 25  &   $P^0_1 \sqrt{2}\, Im(Y^2_2)$ &  $\frac{3}{5} \, Im(  H^L_\perp D^{L\ast}_\parallel +  D^L_\perp H^{L\ast}_\parallel )  $  & $+1$ \\ \hline
 26  &   $P^0_2 \sqrt{2}\, Im(Y^2_2)$ &  $ \sqrt{3} \left[\frac{1}{7} \, Im(D^L_\perp D^{L\ast}_\parallel)   - \frac{1}{5} \, Im(H^L_\perp H^{L\ast}_\parallel)\right] $  & $+1$ \\ \hline
 27  &   $P^0_3 \sqrt{2}\, Im(Y^2_2)$ &  $-\frac{3}{5} \sqrt{ \frac{3}{7}}  \, Im(D^L_\perp H^{L\ast}_\parallel + H^L_\perp D^{L\ast}_\parallel)  $  & $+1$ \\ \hline
 28  &   $P^0_4 \sqrt{2}\, Im(Y^2_2)$ &  $-\frac{4}{7} \sqrt{\frac{3}{5}}  \, Im(D^L_\perp D^{L\ast}_\parallel) $   & $+1$ \\ \hline \hline
 29  &   $P^0_0 Y^0_1$     &  $-\sqrt{3}\left[ \, Re(H^L_\perp H^{L\ast}_\parallel) + \, Re(D^L_\perp D^{L\ast}_\parallel) \right]$  & $-1$ \\ \hline
 30  &   $P^0_1 Y^0_1$     &  $-\frac{3}{\sqrt{5}} \, Re( H^L_\perp D^{L\ast}_\parallel + H^L_\parallel D^{L\ast}_\perp ) $ & $-1$ \\ \hline
 31  &   $P^0_2 Y^0_1$     &  $-\frac{3}{\sqrt{15}} \left[ \frac{5}{7} \, Re(D^L_\perp D^{L\ast}_\parallel) - \, Re(H^L_\perp H^{L\ast}_\parallel)  \right]$ & $-1$ \\ \hline
 32  &   $P^0_3 Y^0_1$     &  $\frac{9}{\sqrt{105}}  \, Re(H^L_\perp D^{L\ast}_\parallel  + H^L_\parallel D^{L\ast}_\perp ) $ & $-1$ \\ \hline
 33  &   $P^0_4 Y^0_1$     &  $\frac{4\sqrt{3}}{7} \, Re(D^L_\perp D^{L\ast}_\parallel)$  & $-1$ \\ \hline
 34  &   $P^1_1 \sqrt{2}\, Re(Y^1_1)$   & $\sqrt{\frac{3}{5}} \left[ \sqrt{5} \, Re(H^L_\perp S^{L \ast})  + \sqrt{3} \, Re(D^L_\perp H^{L\ast}_0)  - \, Re(H^L_\perp D^{L \ast}_0) \right]$  & $-1$ \\ \hline
 35  &   $P^1_2 \sqrt{2}\, Re(Y^1_1)$   & $ 3 \left[ \frac{1}{\sqrt{5}} \, Re(H^L_\perp H^{L \ast}_0)  + \frac{1}{\sqrt{3}} \, Re(D^L_\perp S^{L\ast})  + \frac{5}{21} \sqrt{\frac{3}{5}} \, Re(D^L_\perp D^{L \ast}_0 ) \right] $  & $-1$ \\ \hline
 36  &   $P^1_3 \sqrt{2}\, Re(Y^1_1)$   & $ \frac{6}{\sqrt{70}} \left[ 2 \, Re(D^L_\perp H^{L \ast}_0)  + \sqrt{3} \, Re(H^L_\perp D^{L\ast}_0) \right]$  & $-1$ \\ \hline
 37  &   $P^1_4 \sqrt{2}\, Re(Y^1_1)$   & $\frac{3 \sqrt{10}}{7} \, Re(D^L_\perp D^{L \ast}_0 ) $  & $-1$ \\ \hline
 38  &   $P^1_1 \sqrt{2}\, Im(Y^1_1)$   & $-\sqrt{\frac{3}{5}} \left[ \sqrt{5} \, Im ( H^L_\parallel S^{L\ast}) + \sqrt{3} \, Im(D^L_\parallel H^{L \ast}_0) - \, Im(H^L_\parallel D^{L \ast}_0)  \right]  $  & $-1$ \\ \hline
 39  &   $P^1_2 \sqrt{2}\, Im(Y^1_1)$   & $ -\sqrt{\frac{3}{5}} \left[ \sqrt{3} \, Im(H^L_\parallel H^{L \ast}_0)  + \sqrt{5} \, Im(D^L_\parallel S^{L\ast})  + \frac{5}{7} \, Im(D^L_\parallel D^{L \ast}_0 )\right] $  & $-1$ \\ \hline
 40  &   $P^1_3 \sqrt{2}\, Im(Y^1_1)$   & $ -6\sqrt{\frac{1}{70}}\left[ 2\, Im(D^L_\parallel H^{L \ast}_0)  + \sqrt{3} \, Im(H^L_\parallel D^{L\ast}_0)\right]$   & $-1$ \\ \hline
 41  &   $P^1_4 \sqrt{2}\, Im(Y^1_1)$   & $-\frac{3}{7} \sqrt{10} \, Im(D^L_\parallel D^{L\ast}_0) $  & $-1$ \\ \hline
\end{tabular}
}
\end{document}
