Let $g(m)$ be the integer defined by the following double sum of products of binomial coefficients:

$$\sum _{j=0}^m\sum _{i=0}^j(-1{)}^{j-i}(\genfrac{}{}{0ex}{}{m}{j})(\genfrac{}{}{0ex}{}{j}{i})(\genfrac{}{}{0ex}{}{j+5+6i}{j+5}).$$

You are given that $g(10)=127278262644918$.
Its first (most significant) five digits are $12727$.
Find the first ten digits of $g(142857)$ when written in base $7$.