Say $F(X) \in \mathbb{Z}[X]$ is an even degree polynomial of degree $2n$.

One needs to evaluate $F(X)$ at $O(n)$ points to interpolate and get all the coefficients of $F(X)$.

However say I need only the coefficient of $X^{2n}$, do I still have to evaluate at $O(2n)$ points?

Will having coefficient of $X^{t}$ same as coefficient of $X^{2n-t}$ help in reducing the number of points from $O(n)$?

Is this a well studied problem that has some good references - that is interpolating for only one or few coefficients?

There is one way to do this - evaluating at one large prime and reduction via modulo operations. However, this gives way too much information(that is I can get all the coefficients) and when I evaluate at a large prime, the word size become the order of $O(n\log(nM))$ where M is the largest coefficient size. So in a way we are still using $O(n^{1+\epsilon})$ operations.

I am guessing there should be a way to get only information about the single coefficient I am interested in while getting the operations down to $O(n^{1-\epsilon})$ at the 'cost of not-getting' information about other coefficients.