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This work is a first step towards a theory of "$q$-deformed complex numbers".Assuming the invariance of the $q$-deformation under the action of the modulargroup I prove the existence and uniqueness of the operator of translationsby~$i$ compatible with this action. Obtained in such a way $q$-deformedGaussian integers have interesting properties and are related to the Chebyshevpolynomials.

Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group