Binary quadratic form representing an integer with a vector of level N

Question

Let $N>1$ be an integer. Say I have a binary quadratic form $$ M = \begin{pmatrix} a & b/2\\b/2 & Nc \end{pmatrix} $$ with $a,b,c$ integers which is not positive semidefinite. Then I know there is a vector $g=(x,y)\in\mathbb{Z}^2$ with coprime entries such that $$ g^T M g < 0. $$ Can we always find a vector $h = (s,Nt)\in\mathbb{Z}^2$ with $\gcd(s,Nt)=1$ and $h^TMh<0$? If not, is there always a vector $h=(s,t)\in\mathbb{Z}^2$ with $\gcd(s,t) = \gcd(s,N) = \gcd(t,N) = 1$?