Calculate a table of index vectors for the case $m =40$

Question

I need help calculating a table of index vectors when $m =40$. I also want to know how to use the table to solve $13x^2 \equiv 21 \bmod 40$. Any help is appreicated

Answer 1

Let's first ask for $13y \equiv 21 \bmod 40$, we'll find $y=17+40n$, a number that always ends with $7$, something squares never do...

PS: Sorry I don't know what you mean by "table of index vectors".

Answer 2

$13x^2 \equiv 21 \bmod 40$ reduces to $3x^2 \equiv 1 \bmod 5$ or $x^2 \equiv 2 \bmod 5$, which has no solutions.