Can You Hear the Shape of a Lattice?

After Lecture 23, Emanuele Viola asked me whether or not the following is true: ``If $f_1$ and $f_2$ are binary quadratic forms that represent exactly the same integers, is $f_1\sim f_2$?'' The answer is no. For example, $f_1=(2,1,3)=2x^2 + xy+3y^2$ and $f_2=(2,-1,3)=2x^2 -xy+3y^2$ are inequivalent reduced positive definite binary quadratic forms that represent exactly the same integers. Note that $\disc(f_1) = \disc(f_2)=-23$. There appears to be a sense in which all counterexamples resemble the one just given.

Questions like these are central to John H. Conway's book The sensual (quadratic) form, which I've never seen because the Cabot library copy is checked out and the Birkhoff copy has gone missing. The following is taken from the MATHSCINET review (I changed the text slightly so that it makes sense):

Chapter $2$ begins by posing Mark Kac's question of ``hearing the shape of a drum'', and the author relates the higher-dimensional analogue of this idea on tori--quotients of $\mathbf{R}\sp n$ by a lattice--to the question of what properties of a positive definite integral quadratic form are determined by the numbers the form represents. A property of such a form is called ``audible'' if the property is determined by these numbers, or equivalently, by the theta function of the quadratic form. As examples, he shows that the determinant of the form and the theta function of the dual form are audible. He also provides counterexamples to the higher-dimensional Kac question, the first of which were found by J. Milnor...