  Let $p>3$ be a prime and $(\frac{.}{p})$ be the Legendre symbol. For any
integer $d$ with $p\nmid d$ and any positive integer $m$, Sun introduced the
determinants
$$T_m(d,p)=\det\left[(i^2+dj^2)^m\left(\frac{i^2+dj^2}{p}\right)\right]_{1\leqslant
i,j \leqslant (p-1)/2},$$ and $$D_p^{(m)}=
\det\left[(i^2-j^2)^m\left(\frac{i^2-j^2}{p}\right)\right]_{1\leq i,j\leq
(p-1)/2} .$$ In this paper, we obtain some properties of $T_m (d,p)$ and $
\sqrt{D_p^{(m)}}$ for some $m$. We also confirm some related conjectures posed
by Zhi-Wei Sun.
