  In this article, we prove a new bound for the least prime ideal in the
Chebotarev density theorem, which improves the main theorem of Zaman [Funct.
Approx. Comment. Math. 57 (2017), no.1, 115-142] by a factor of $5/2$. Our main
improvement comes from a new version of Tur\'an's power sum method. The key new
idea is to use Harnack's inequality for harmonic functions to derive a superior
lower bound for the generalised Fej\'er kernel.
