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We present stellar population parameters of 12 elliptical and S0 galaxies in the Coma cluster around and including the cD galaxy NGC 4874, based on spectra obtained using the Low Resolution Imaging Spectrograph on the Keck II Telescope. Our data are among the most precise and accurate absorption-line strengths yet obtained for cluster galaxies, allowing us to examine in detail the zero-point and scatter in the stellar population properties of Coma cluster early-type galaxies (ETGs). Recent observations of red sequence galaxies in the high-redshift Universe and generic hierarchical galaxy formation models lead to the following expectations for the stellar populations of local ETGs. (1) In all environments, bigger ETGs should have older stellar populations than smaller ETGs (`downsizing'); (2) ETGs at fixed stellar mass form stars earlier and thus should have older stellar population ages in the highest density environments than those in lower density environments and (3) the most massive ETGs in the densest environments should have a small spread in stellar population ages. We find the following surprising results using our sample. (1) Our ETGs have single-stellar-population-equivalent (SSP-equivalent) ages of on average 5-8 Gyr with the models used here, with the oldest galaxies having SSP-equivalent ages of ≲10-Gyr old. This average age is identical to the mean age of field ETGs. (2) The ETGs in our sample span a large range in velocity dispersion (mass) but are consistent with being drawn from a population with a single age. Specifically, 10 of the 12 ETGs in our sample are consistent within their formal errors of having the same SSP-equivalent age, 5.2±0.2 Gyr, over a factor of more than 750 in mass. We therefore find no evidence for downsizing of the stellar populations of ETGs in the core of the Coma cluster. We confirm the lack of a trend of SSP-equivalent age with mass in the core of the Coma cluster from all other samples of Coma cluster ETG absorption-line strengths available in the literature, but we do find from the largest samples that the dispersion in age increases with decreasing mass. These conclusions stand in stark contrast to the expectations from observations of high-redshift red sequence galaxies and model predictions. We suggest that Coma cluster ETGs may have formed the majority of their mass at high redshifts but suffered small but detectable star formation events at z ∼ 0.1-0.3. In this case, previous detections of `downsizing' from stellar populations of local ETGs may not reflect the same downsizing seen in look-back studies of RSGs, as the young ages of the local ETGs represent only a small fraction of their total masses.