The number of representations of integers by ternary subforms of x2 + y2 + z2 or x2 + y2+ 2z2

In this article, we find all spaces of cusp forms of weight 32 and dimension 1. Furthermore, we construct bases for those spaces consisting of eta-quotients and find exact formulas for their Fourier coefficients. As applications, we provide closed formulas for the number of representations of integers by ternary subforms of x2 + y2 + z2 or x2 + y2 + 2z2 (see Table 1). (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.