This paper studies a delicate arithmetic question: how the coefficients of symmetric-power L-functions behave when you look only at numbers that can be written as a sum of m squares and lie in a fixed arithmetic progression. The coefficients in question, written λ_{sym^j f}(n), come from taking the j-th symmetric power of the L-function attached to a classical Hecke eigenform f. The authors obtain upper bounds for partial sums and second moments of these coefficients over those special integers. They then use these bounds to control shifted convolution sums that include a k‑full kernel, and to count sign changes of the twisted coefficients.

Concretely, the paper fixes an even m from the list {2,4,6,8,10,12} and a fixed symmetric power index j≥2. It examines sums over integers n that are sums of m squares and are congruent to 1 modulo q. Under assumptions such as q≥100 and q not too large compared with the summation range x (made precise in the statements), the authors prove explicit big‑O upper bounds for the sum of λ_{sym^j f}(n) and for its second moment. They also factor auxiliary Dirichlet series built from these coefficients. Those factorizations express the series as products of twisted symmetric-power L-functions times a remaining Dirichlet series that converges in a half‑plane, and they use these analytic properties to get the arithmetic estimates.

As an application, the paper treats shifted convolution sums where the coefficients are weighted by a k‑full kernel function a(n). A k‑full number is one where every prime divisor occurs to power at least k. The k‑full kernel functions used are a standard class introduced by Ivić and Tenenbaum and can be non-multiplicative. For any integer k≥2 the authors give power-saving upper bounds for sums of the form ∑ a(n) λ_{sym^j f}(n+1) and for similar second moments, with error terms that improve as k grows. These bounds are proved for the same types of summation sets (sums of m squares, congruent conditions) and again under ranges on q relative to x.