  This work builds on earlier results. We define universal elliptic Gau{\ss}
sums for Atkin primes in Schoof's algorithm for counting points on elliptic
curves. Subsequently, we show these quantities admit an efficiently computable
representation in terms of the $j$-invariant and two other modular functions.
We analyse the necessary computations in detail and derive an alternative
approach for determining the trace of the Frobenius homomorphism for Atkin
primes using these pre-computations. A rough run-time analysis shows, however,
that this new method is not competitive with existing ones.
