Let $I$ be an ideal of $\mathbb{C}[GL_n]$. Are there effective methods or software to check whether $I$ is a coideal or not? Thank you very much.

For example, let I be the ideal of $\mathbb{C}[GL_3]$ generated by the following elements (the indices start from $0$):
\begin{align}
& c_{1,2}  {c_{0,0}}^2 - 2  c_{1,1}  c_{0,0}  c_{0,1} - c_{1,0}  c_{0,2}  c_{0,0} + 2  c_{1,0}  {c_{0,1}}^2, \\ 
& c_{0,0}  c_{0,1}  c_{1,2} - 2  c_{0,0}  c_{0,2}  c_{1,1} + c_{0,1}  c_{1,0}  c_{0,2}, \\ 
& 2  c_{2,1}  {c_{0,0}}^2 - c_{1,1}  c_{0,0}  c_{1,0} - 2  c_{0,1}  c_{2,0}  c_{0,0} + c_{0,1}  {c_{1,0}}^2, \\ 
& c_{2,2}  {c_{0,0}}^2 - c_{0,0}  {c_{1,1}}^2 - c_{0,2}  c_{2,0}  c_{0,0} + c_{0,1}  c_{1,0}  c_{1,1}, \\ 
& 2  c_{0,0}  c_{0,1}  c_{2,2} - 2  c_{0,0}  c_{0,2}  c_{2,1} - c_{0,0}  c_{1,1}  c_{1,2} + c_{0,1}  c_{1,0}  c_{1,2}, \\ 
& 2  c_{1,2}  {c_{0,1}}^2 - 2  c_{1,1}  c_{0,1}  c_{0,2} + c_{1,0}  {c_{0,2}}^2 - c_{0,0}  c_{1,2}  c_{0,2}, \\ 
&  - 4  c_{2,0}  {c_{0,1}}^2 + 4  c_{0,0}  c_{2,1}  c_{0,1} + c_{0,2}  {c_{1,0}}^2 - c_{0,0}  c_{1,2}  c_{1,0}, \\ 
& 2  c_{0,0}  c_{0,1}  c_{2,2} - c_{0,0}  c_{1,1}  c_{1,2} - 2  c_{0,1}  c_{0,2}  c_{2,0} + c_{1,0}  c_{0,2}  c_{1,1}, \\ 
& 4  c_{2,2}  {c_{0,1}}^2 - 4  c_{0,2}  c_{2,1}  c_{0,1} - c_{0,0}  {c_{1,2}}^2 + c_{1,0}  c_{0,2}  c_{1,2}, \\ 
& c_{0,0}  c_{1,0}  c_{2,1} - 2  c_{0,0}  c_{1,1}  c_{2,0} + c_{0,1}  c_{1,0}  c_{2,0}, \\ 
& c_{0,0}  c_{1,0}  c_{2,2} - 2  c_{0,0}  c_{1,1}  c_{2,1} - c_{0,0}  c_{2,0}  c_{1,2} + 2  c_{0,1}  c_{1,0}  c_{2,1}, \\ 
& c_{0,1}  c_{1,0}  c_{2,2} - c_{0,0}  c_{1,2}  c_{2,1}, \\ 
& c_{0,0}  c_{1,0}  c_{2,2} - 2  c_{0,0}  c_{1,1}  c_{2,1} + 2  c_{0,1}  c_{1,1}  c_{2,0} - c_{1,0}  c_{0,2}  c_{2,0}, \\ 
& c_{1,0}  c_{0,2}  c_{2,1} - c_{0,1}  c_{2,0}  c_{1,2}, \\ 
& 2  c_{0,1}  c_{1,1}  c_{2,2} - c_{0,0}  c_{1,2}  c_{2,2} - 2  c_{0,1}  c_{1,2}  c_{2,1} + c_{1,0}  c_{0,2}  c_{2,2}, \\ 
& c_{2,1}  {c_{1,0}}^2 - c_{1,1}  c_{1,0}  c_{2,0} + 2  c_{0,1}  {c_{2,0}}^2 - 2  c_{0,0}  c_{2,1}  c_{2,0}, \\ 
& c_{2,2}  {c_{1,0}}^2 - c_{2,0}  c_{1,2}  c_{1,0} - 4  c_{0,0}  {c_{2,1}}^2 + 4  c_{0,1}  c_{2,0}  c_{2,1}, \\ 
& 2  c_{0,1}  c_{2,0}  c_{2,2} - 2  c_{0,0}  c_{2,1}  c_{2,2} + c_{1,0}  c_{1,1}  c_{2,2} - c_{1,0}  c_{1,2}  c_{2,1}, \\ 
& 2  c_{0,0}  c_{0,2}  c_{2,1} - c_{0,1}  c_{1,0}  c_{1,2} - 2  c_{0,1}  c_{0,2}  c_{2,0} + c_{1,0}  c_{0,2}  c_{1,1}, \\ 
&  - c_{2,0}  {c_{0,2}}^2 + c_{0,2}  {c_{1,1}}^2 + c_{0,0}  c_{2,2}  c_{0,2} - c_{0,1}  c_{1,2}  c_{1,1}, \\ 
&  - 2  c_{2,1}  {c_{0,2}}^2 + c_{1,1}  c_{0,2}  c_{1,2} + 2  c_{0,1}  c_{2,2}  c_{0,2} - c_{0,1}  {c_{1,2}}^2, \\ 
& c_{0,0}  c_{2,0}  c_{1,2} - 2  c_{0,1}  c_{1,0}  c_{2,1} + 2  c_{0,1}  c_{1,1}  c_{2,0} - c_{1,0}  c_{0,2}  c_{2,0}, \\ 
& c_{0,0}  c_{1,2}  c_{2,2} - 2  c_{0,1}  c_{1,1}  c_{2,2} + 2  c_{0,2}  c_{1,1}  c_{2,1} - c_{0,2}  c_{2,0}  c_{1,2}, \\ 
& 2  c_{0,2}  c_{1,1}  c_{2,2} - c_{0,1}  c_{1,2}  c_{2,2} - c_{0,2}  c_{1,2}  c_{2,1}, \\ 
&  - {c_{1,1}}^2  c_{2,0} + c_{1,0}  c_{2,1}  c_{1,1} + c_{0,2}  {c_{2,0}}^2 - c_{0,0}  c_{2,2}  c_{2,0}, \\ 
& c_{1,0}  c_{1,1}  c_{2,2} - 2  c_{0,0}  c_{2,1}  c_{2,2} + 2  c_{0,2}  c_{2,0}  c_{2,1} - c_{1,1}  c_{2,0}  c_{1,2}, \\ 
& {c_{1,1}}^2  c_{2,2} - c_{1,2}  c_{2,1}  c_{1,1} - c_{0,0}  {c_{2,2}}^2 + c_{0,2}  c_{2,0}  c_{2,2}, \\ 
& 2  c_{0,1}  c_{1,2}  c_{2,1} - c_{1,0}  c_{0,2}  c_{2,2} - 2  c_{0,2}  c_{1,1}  c_{2,1} + c_{0,2}  c_{2,0}  c_{1,2}, \\ 
& c_{1,0}  c_{1,2}  c_{2,1} - 2  c_{0,1}  c_{2,0}  c_{2,2} + 2  c_{0,2}  c_{2,0}  c_{2,1} - c_{1,1}  c_{2,0}  c_{1,2}, \\ 
&  - c_{2,0}  {c_{1,2}}^2 + c_{1,0}  c_{2,2}  c_{1,2} + 4  c_{0,2}  {c_{2,1}}^2 - 4  c_{0,1}  c_{2,2}  c_{2,1}, \\ 
&  - c_{2,1}  {c_{1,2}}^2 + c_{1,1}  c_{1,2}  c_{2,2} - 2  c_{0,1}  {c_{2,2}}^2 + 2  c_{0,2}  c_{2,1}  c_{2,2}, \\ 
&  - c_{1,2}  {c_{2,0}}^2 + 2  c_{1,1}  c_{2,0}  c_{2,1} + c_{1,0}  c_{2,2}  c_{2,0} - 2  c_{1,0}  {c_{2,1}}^2, \\ 
& 2  c_{1,1}  c_{2,0}  c_{2,2} - c_{1,0}  c_{2,1}  c_{2,2} - c_{2,0}  c_{1,2}  c_{2,1}, \\ 
&  - 2  c_{1,2}  {c_{2,1}}^2 + 2  c_{1,1}  c_{2,1}  c_{2,2} - c_{1,0}  {c_{2,2}}^2 + c_{2,0}  c_{1,2}  c_{2,2}.
\end{align}