Digit Sums Across Bases

For a positive integer $n$, let $f(n)$ be the digit sum of $n$ written in base $4$, and $g(n)$ the digit sum of $f(n)$ written in base $8$. (Example: $f(2020) = 10 = 12_{(8)}$, so $g(2020) = 3$.) Let $N$ be the least $n$ for which the base-$16$ representation of $g(n)$ requires a digit beyond $0$–$9$. Find $N \bmod 1000$.