The minimal positive integer represented by a positive definite quadratic form

Abstract.

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 $\Gamma_{0}(2)$. 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 $\mathcal{Q}$ be an even positive definite quadratic form of level two in $v$ variables, with $v \equiv 4(\textrm{mod}\, 8)$, then $\mathcal{Q}$ represents a positive integer $2n \leq 3+v/4$.