The eligible numbers of positive definite ternary forms

Several nontrivial genera of positive ternary forms with small discriminant s have been studied in this paper. Especially we prove that there are only finitely many, square-free eligible numbers in the genus of f(1) = x(2) + y (2) + 7z(2) and in the genus of f(2) = x(2) + 7y(2) + 7x(2) which cannot be represented by f(1) and f(2), respectively. Our method is to use modular f orms of weight 3/2.