Given the equation $$-4 \left(a^2+a (n-1) (2 t^2-1)\right) \left(\sum _{i=0}^n \alpha_i t^{2 i}\right)^2 \\ +\frac12 \left(\sum _{i=0}^n \alpha_i t^{2 i}\right) \left(t \left(8 a (t^2-1)+1\right) \sum _{i=0}^n 2 i \alpha_i t^{2 i-1}+(t^2-1) \sum _{i=0}^n 2 i (2 i-1) \alpha_i t^{2 i-2}\right) \\ -\frac14(t^2-1) \left(\sum _{i=0}^n 2 i \alpha_i t^{2 i-1}\right)^2=4C\left(\sum _{i=0}^n \alpha_i t^{2 i}\right)^2,$$
where $t \in[-1,1]$, $a$ being a fixed in $\mathbb{R}$ and $C,\alpha_0,...,\alpha_n$ unknown. I am looking for a way to numerically compute the unknown variables. In principle, this is very similar to an eigenvalue problem, where $C$ is the eigenvalue. The problem is that the equation is not linear in the $\alpha_i$. What would be the best way to solve this equation numerically?