We employ a set of Magneticum cosmological hydrodynamic simulations that span over 15 different cosmologies, and extract masses and concentrations of all well-resolved haloes between z = 0 and 1 for critical overdensities Δvir, Δ200c, Δ500c, Δ2500c and mean overdensity Δ200m. We provide the first mass-concentration (Mc) relation and sparsity relation (i.e. MΔ1 - MΔ2 mass conversion) of hydrodynamic simulations that is modelled by mass, redshift, and cosmological parameters ωm, ωb, σ8, h0 as a tool for observational studies. We also quantify the impact that the Mc relation scatter and the assumption of Navarro-Frank-White (NFW) density profiles have on the uncertainty of the sparsity relation. We find that converting masses with the aid of an Mc relation carries an additional fractional scatter (4 percent;) originated from deviations from the assumed NFW density profile. For this reason, we provide a direct mass-mass conversion relation fit that depends on redshift and cosmological parameters. We release the package hydro_mc, a python tool that perform all kind of conversions presented in this paper.
