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].