# Unlucky Polynomial

**Algebra**Level 5

Suppose \(f(x)\) is a polynomial of degree \(13\) that satisfies the functional equation \[f(2x^2-1)=2(f(x))^2-1\] Find the last three digits of the coefficient of \(x^5\) in \(f\).

**Details and assumptions**

The coefficient of \(x^5\) is a positive integer.