I am wondering is there a nice relation between the number of representations of a positive, primitive, integral ternary form and its number of positive automorphisms? More explicitly, let $f=ax^2+by^2+cz^2+ryz+txz+sxy$ be a positive integral ternary form with the gcd of its coefficients is 1. Let $R_n(f)=\{v=(\alpha,\beta,\gamma)\in\mathbb{Z}: f(\alpha,\beta,\gamma)=n, \gcd(\alpha,\beta,\gamma)=1\}.$ Let $Aut^+(f)=\{M\in SL_3(\mathbb{Z}): fM=f\}.$
My question is that is there a relation between $|Aut^+(f)|$ and $|R_n(f)|.$ I analyzed an action of $Aut^+(f)$ on $R_n(f),$ however, I couldn't get a result. Also, I constructed an example and saw that there is a form $f$, and there is a nonidentity $M\in Aut^+(f)$ such that $Mv_1=v_1$ for $v_1\in R_4(f)$. Hence we cannot say any nonidentity automorph gives a different representation.
Lastly, in the binary case, we have a very nice relation. That's why I also believe that there may be a nice one in the ternary case as well.