We present VLT/SINFONI observations of 35 quasars at 2.1 < z < 3.2, the majority of which were selected from the Clusters Around Radio-Loud AGN (CARLA) survey. CARLA quasars have large CIV-based black hole (BH) masses (M(BH) > 10^9 Msun) and powerful radio emission (P(500MHz) > 27.5 W/Hz). We estimate Ha-based M(BH), finding a scatter of 0.35 dex compared to CIV. We evaluate several recipes for correcting CIV-based masses, which reduce the scatter to 0.24 dex. The radio power of the radio-loud quasars is at most weakly correlated with the interconnected quantities Ha-width, L(5100A) and M(BH), suggesting that it is governed by different physical processes. However, we do find a strong inverse correlation between CIV blueshift and radio power linked to higher Eddington ratios and L(5100A). Under standard assumptions, the BH growth time is longer than the cosmic age for many CARLA quasars, suggesting that they must have experienced more efficient growth in the past. If these BHs were growing from seeds since the epoch of reionization, it is possible that they grew at the Eddington limit like the quasars at z ~ 6-7, and then continued to grow at the reduced rates observed until z ~ 2-3. Finally, we study the relation between M(BH) and environment, finding a weak positive correlation between M(BH) and galaxy density measured by CARLA.
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