Spinor genera of binary quadratic forms

Spinor genera are defined for binary quadratic forms with integer coefficients in such a way that the theory fits in with the Gaussian theory of genera. It is shown that spinor generic characters exist which distinguish the various spinor genera in the principal genus, and how they can be determined. It is known that each ambiguous class contains exactly two forms of the type [a, 0, c] or [a, a, c], each with its associate [c, 0, a], [4c ? a, 4c ? a, c]. Since the principal class contains such a form with a = 1, it is an interesting question whether one can predict the second form (not counting associates). This question includes that of Dirichlet about the representability of ?1 by the principal class. Methods are given for evaluating the spinor-generic characters of ambiguous forms in the principal genus for variable discriminants d, and are carried through in the eleven cases where d is fundamental, there are two or four genera, and two spinor genera in the principal genus. The problem of determining the “second form” is thus completely solved except when there is more than one ambiguous class in the principal spinor genus.