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When $n$ is odd there is no alternating sign mode. The same cosine-sine construction gives invariant irreducible planes $W_k$ foragain pairwise nonisomorphic because their generator traces are distinct. The real Fourier decomposition isIts dimensions sum to $1+2(n-1)/2=n$. Thus the real regular representation of a finite cyclic group has one one-dimensional irreducible summand and $(n-1)/2$ nonisomorphic two-dimensional irreducible summands.