The classification and representations of ternary quadratic forms with level $ 8N $

In this paper, we investigated positive definite ternary quadratic forms of level $ 8N $, where $ N $ is an odd positive squarefree integer. Our study makes two main contributions. First, we provided an explicit classification of positive definite ternary quadratic forms of level $ 8N $. Second, we derived exact formulas for the weighted sum of representations over each class within every genus of ternary quadratic forms of level $ 8N $, which involved modified Hurwitz class numbers. The proof of our main results leverages the relations between ternary quadratic forms, quaternion algebras, and weight $ 3/2 $ modular forms of level $ 8N $. As applications, we obtained exact formulas for the class number of positive ternary quadratic forms of level $ 8N $.