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We report our survey of $\gamma$\,Doradus stars from the 4-yr \textit{Kepler}mission. These stars pulsate mainly in g modes and r modes, showingperiod-spacing patterns in the amplitude spectra. The period-spacing patternsare sensitive to the chemical composition gradients and the near-core rotation,hence they are essential for understanding the stellar interior. We identifiedperiod-spacing patterns in 611 $\gamma$\,Dor stars. Almost every star pulsatesin dipole g modes, while about 30\% of stars also show clear patterns forquadrupole g modes and 16\% of stars present r mode patterns. We measureperiods, period spacings, and the gradient of the period spacings. These threeobservables guide the mode identifications and can be used to estimate thenear-core rotation rate. We find many stars are hotter and show longerperiod-spacing patterns than theory. Using the Traditional Approximation ofRotation (TAR), we inferred the asymptotic spacings, the near-core rotationrates, and the radial orders of the g and r modes. Most stars have a near-corerotation rate around $1$\,$\mathrm{c/d}$ and an asymptotic spacing around4000\,s. We also find that many stars rotate more slowly than predicted bytheory for unclear reasons. 11 stars show rotational splittings with fastrotation rates. We compared the observed slope--rotation relation with thetheory and find a large spread. We detected rotational modulations in 58 starsand used them to derive the core-to-surface rotation ratios. The interiorsrotate faster than the cores in most stars, but by no more than 5\%.

Gravity-mode period spacings and near-core rotation rates of 611 $\gamma$ Doradus stars with \textit{Kepler}