On certain determinants and the square root of some determinants involving Legendre Symbols

Let p>3 be a prime and (. p) be the Legendre symbol. For any integer d with p d and any positive integer m, Sun introduced the determinants Tₘ (d, p) = (i²+dj²) ᵐ (i²+dj²p) ₁ ₈, ₉ (₏-₁) /₂, and Dₚ^ (m) = (i²-j²) ᵐ (i²-j²p) ₁ ₈, ₉ (₏-₁) /₂. In this paper, we obtain some properties of Tₘ (d, p) and Dₚ^ (m) for some m. We also confirm some related conjectures posed by Zhi-Wei Sun.