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We present a new extended and detailed set of models for Classical Cepheidpulsators at solar chemical composition ($Z=0.02$, $Y=0.28$) based on a welltested nonlinear hydrodynamical approach. In order to model the possibledependence on crucial assumptions such as the Mass-Luminosity relation ofcentral Helium burning intermediate-mass stars or the efficiency ofsuperadiabatic convection, the model set was computed by varying not only thepulsation mode and the stellar mass but also the Mass-Luminosity relation andthe mixing length parameter that is used to close the system of nonlinearhydrodynamical and convective equations. The dependence of the predictedboundaries of the instability strip as well as of both light and radialvelocity curves on the assumed Mass-Luminosity and the efficiency ofsuperadiabatic convection is discussed. NonlinearPeriod-Mass-Luminosity-Temperature, Period-Radius and Period-Mass-Radiusrelations are also computed. The theoretical atlas of bolometric light curvesfor both the fundamental and first overtone mode has been converted in the Gaiafilters $G$, $G_{BP}$ and $G_{BR}$ and the corresponding mean magnitudes havebeen derived. Finally the first theoretical Period-Luminosity-Color andPeriod-Wesenheit relations in the Gaia filters are provided and the resultingtheoretical parallaxes are compared with Gaia Data Release 2 results for bothfundamental and first overtone Galactic Cepheids.

An extended theoretical scenario for Classical Cepheids. I. Modeling Galactic Cepheids in the Gaia photometric system