Diagonal quinary quadratic forms with a strong regularity property

Let f be a positive definite integral quinary 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 19 strongly s-regular diagonal quinary quadratic forms representing 1 (see Table 1). In particular, we use etaquotients of weight 52 to prove the strongly s-regularity of the quinary quadratic form x2 + 2y2 + 2z2 + 3w2 + 3t2, which is, in fact, of class number 4. (c) 2022 Elsevier Inc. All rights reserved.