Introduction

The following computer-generated tables of reduced regular primitive positive definite quaternary quadratic forms over the rational integers are inspired by and are an outgrowth of the remarkable Brandt-Intrau tables of reduced positive ternary forms [2], published in pre-computer days after what must have been an incredible amount of effort. The Brandt-Intrau tables serve not only as a model of accuracy but also as a starting point for generating classes of reduced forms with an additional variable by computer. The technique used for this was essentially that of Germann [13], who began with the ternary forms through discriminant 16 from the Brandt-Intrau tables and computed all classes of reduced positive quaternary forms through discriminant 61. While Germann had few enough forms to allow separation into classes on an individual basis, our task was somewhat less manageable. Beginning with the ternary forms of discriminant δ ≤ 320 from the Brandt-Intrau tables (with the imprimitive ones adjoined), we generated a large file of quaternary forms containing representatives of all possible classes. Directed by a conjecture of John Hsia [14] that no two inequivalent such forms with the same discriminant would have the same theta series, two forms with the same discriminant were considered “equivalent” as a preliminary sorting measure if their first twenty theta coefficients were the same (the number 20 being chosen arbitrarily).