''Assume that the generalized Riemann hypothesis (GRH) for zeta functions of number ﬁelds holds. There exists a deterministic algorithm that on input positive integers $n$ and $k$, together with the factorization of $n$ into prime factors, computes the element $T_n$ of the Hecke algebra $T(1, k)$ in running time polynomial in $k$ and $\log n$.''