Does anyone know how to easily compute them? We know that a number is a square modulo $2^k$ if and only if it's a square modulo $8$. This gives a bunch of integers that represent square classes. I also know that
$\mathbb{Q}_2^\times \cong \mathbb{Z}\times (1+2\mathbb{Z}_2),$
but I can't figure out how to find the square classes in $1+2\mathbb{Z}_2$. Is there some really simple solution for this? I can't seem to figure out how to apply Hensel's lemma here.