The minimal positive integer represented by a positive definite quadratic form

In this paper we shall give an upper bound on the size of the gap between the constant term and the next nonzero Fourier coefficient of a holomorphic modular form of given weight for the group G₀(2) \Gamma_{0}(2) . We derive an upper bound for the minimal positive integer represented by an even positive definite quadratic form of level two. In our paper we prove two conjectures given in 1]. In particular, we can prove the following result: let Q \mathcal{Q} be an even positive definite quadratic form of level two in v v variables, with v o 4(mod 8) v \equiv 4(\textrm{mod}\, 8) , then Q \mathcal{Q} represents a positive integer 2n ￡ 3+v/4 2n \leq 3+v/4 .