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We consider smoothness properties of the generator of a principal Gabor spaceon the real line which is invariant under some additionaltranslation-modulation pair. We prove that if a Gabor system on a lattice withrational density is a Riesz basis for its closed linear span, and if the closedlinear span, a Gabor space, has any additional translation-modulationinvariance, then its generator cannot decay well in time and in frequencysimultaneously.

Time-frequency shift invariance and the Amalgan Balian Low Theorem