HMMT 2025 A9

Let $f$ be the unique polynomial of degree at most 2026 such that for all $n\in \{1,2,3,\ldots ,2027\}$

Suppose that $\frac{a}{b}$ is the coefficient of $x^{2025}$ in $f$ , where $a$ and $b$ are integers such that $\gcd (a,b) = 1$ . Compute the unique integer $r$ between 0 and 2026 (inclusive) such that $a - rb$ is divisible by 2027. (Note that 2027 is prime.)