The structures of neural population activity, such as the dimension, have received much interest thanks to advances in simultaneous recordings of a large number of neurons. An important question is how these structures emerge mechanistically and relate to connectivity. Here we study the spectrum of the population covariance, which is closely related to the linear dimension and PCA. We consider a randomly connected recurrent network with linear dynamics fluctuating around a steady state. The analytically derived spectrum is continuous and finitely supported with a long tail that exhibits a power law as the connection strength increases. The results are then generalized to describe certain E-I networks and the effects of connectivity motifs and temporal and spatial sampling in experimental data. Applications to whole-brain imaging data and nonlinear dynamics will also be discussed.