Diagonal senary quadratic forms with a strong regularity property

Let f be a positive definite integral senary 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 by f. In this article, we show that there exist exactly 8 strongly s-regular diagonal senary quadratic forms representing 1 (see Table 1). Furthermore, we prove the strongly s-regularity of the senary quadratic form x 2 + 2y 2 + 2z 2 + 4w 2 + 2t 2 + 2s 2 + 2ts, which is, in fact, of class number 5.