The number of representations of squares by integral quaternary quadratic forms III

Let [Formula: see text] be a positive definite (non-classic) integral quaternary quadratic form. We say [Formula: see text] is strongly[Formula: see text]-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this article, we show that there are exactly [Formula: see text] strongly [Formula: see text]-regular quaternary quadratic forms representing [Formula: see text] (see Table ??). Together with the results in [2] and [3], we show that there are precisely [Formula: see text] integral strongly [Formula: see text]-regular quaternary quadratic forms that represent one. In particular, we use modular forms to prove the strongly [Formula: see text]-regularity of the quaternary quadratic form [Formula: see text], which is, in fact, of class number [Formula: see text].