Fomenko [8] proved that n≤x λ sym 2 f (n)x 1/2 (log x) 2 .Since then, this sum has been studied by many scholars, including Sankaranarayanan, Ichihara, Tang, Lü and Jiang (see [31,12,16,33,28]). The higher degree cases were considered by Lau and Lü [25] and Tang and Wu [35]. In particular, under the assumption that L(sym j f, s) is automorphic for j ≥ 1, Tang and Wu showed thatwhere δ j is given by [35, (1.14)].On the other hand, Fomenko [9] studied the sum of λ 2 sym 2 f (n). Later, this result was improved by Tang [34]. X. G. He [11] where δ * 2 = 13 17 , δ * 3 = 248 269 and P 2 (t) is a polynomial in t of degree 2. Lately, Sankaranarayanan, Singh and Srinivas [32] showed that n≤x λ 2 sym 3 f (n) = c 1 x + O(x 15 17 +ε ), n≤x λ 2 sym 4 f (n) = c 2 x + O(x 12 13 +ε ).