The number of representations of squares by integral quaternary quadratic forms

Let f be a positive definite (non-classic) integral quaternary quadratic form. We say f is strongly s-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this paper, we show that there are exactly 34 strongly s-regular diagonal quaternary quadratic forms representing 1 (see Table 1). In particular, we use eta-quotients to prove the strong s-regularity of the quaternary quadratic form x(2) + 2y(2) + 3z(2) + 10w(2), which is, in fact, of class number 2 (see Lemma 4.5 and Proposition 4.6).