We give a refined description of the dominant spectrum of a nonlocal operator that models growth and equal mitosis of cells. More precisely, we look at the spectrum in half planes at the right-hand side of the first accumulation point of eigenvalues and give criteria on the weight of weighted \(L^1\) spaces for this spectrum to be made of explicit simple eigenvalues. The method relies on a high-order long-time asymptotic expansion of the solutions to the associated evolution equation obtained in [Zaidi, van Brunt, Wake, Proc. A, R. Soc. Lond., 2015] combined with a Weyl theorem taken from [Mischler, Scher, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 2016].