Two integral quadratic forms1 $f(x,y) = ax^2+bxy+cx^2$ and $g(x,y)=a'x^2+b'xy+c'y^2$, with $a, b, c, a', b', c' \in \mathbb{Z}$, are said to be properly equivalent if $$ f(x,y) = g(\alpha x + \beta y, \gamma x + \delta y) $$ for some $\alpha, \beta, \gamma, \delta \in \mathbb{Z}$ and $\alpha \delta - \beta \gamma = +1$.
The question: Is there some well-known example of an integral quadratic form $(a,b,c) := a x^2+ bx y + c^2,$ which is not properly equivalent to the opposite form $(a,-b,c) = a x^2 - bx y + c^2$?
This question arose while reading section V of Gauss's Disquisitiones Arithmeticae. I have so far failed in finding an example.
1. Some definitions use $2b$ instead of $b$ in the RHS.
$$ 2x^2 + xy + 3 y^2 $$
To get an example with $\Delta + 4$ a square but the form not equivalent to its opposite, it is necessary to use indefinite forms, which are more work.
$$ 9 x^2 + 3xy - 17 y^2 $$
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