Convolutional neural networks (CNNs) provide a generalizable architecture for encoding models that predict neural activity evoked by natural sounds with much greater accuracy than other established models. However, the complexity of CNNs makes it difficult to discern the computational properties that support their improved predictive power. We used a combination of local linear approximation and dimensionality reduction to measure the tuning subspace captured by a CNN. Single-unit data was recorded using microelectrode arrays from primary auditory cortex (A1) of awake, passively listening ferrets during presentation of a large natural sound set. A CNN was fit to the data, replicating approaches from previous work. To measure the tuning subspace, the dynamic spectrotemporal receptive field (dSTRF) was measured as the locally linear filter approximating the input-output relationship of the CNN at each stimulus timepoint. Principal component analysis then reduced this very large number of filters to define a smaller tuning subspace. Typically, 2-10 filters accounted for 90\% of variance in the dSTRFs. The stimulus was projected into the subspace for each neuron, and a new model was fit using only the projected values. The subspace model was able to predict time-varying spike rate nearly as accurately as the full CNN. Sensory responses could be plotted in the subspace, providing a compact model visualization. This analysis revealed a diversity of nonlinear responses, consistent with contrast gain control and emergent invariance to spectrotemporal modulation phase. Within local populations, neurons formed a sparse representation by tiling the tuning subspace. Narrow spiking, putative inhibitory neurons showed distinct tuning that may reflect their position in the cortical circuit. These results demonstrate a conceptual link between CNN and multifilter models and establish a framework for interpretation of deep learning-based encoding models.