We present a new approach to calculating the higher Frobenius-Schur indicators for the simple modules over the Drinfeld double of a finite group. In contrast to the formula by Kashina-Sommerhäuser-Zhu that involves a sum over all group elements satisfying a certain condition, our formula operates on the level of conjugacy classes and character tables. It can be implemented in the computer algebra system GAP, efficiently enough to deal, on a laptop, with symmetric groups up to S_18 (providing further evidence that indicators are non-negative in this case) or simple groups of order up to 2 · 10^8. The approach also allows us to test whether all indicators over the double of a given group are rational , without computing them. Among simple groups of order up to about 5 · 10^11 an inspection yields exactly one example (of order about 5 · 10^9) where irrational indicators occur.