Advances in simultaneous recordings of large numbers of neurons have driven significant interest in structures of neural population activity such as dimensionality. A key question is how these dynamic features arise mechanistically and their relationship to circuit connectivity. It was previously proposed to use the covariance eigenvalue distribution, or spectrum, which can be analytically derived in random recurrent networks, as a robust measure for describe the shape of neural population activity beyond the dimension (Hu \& Sompolinsky 2022). Applications of the theoretical spectrum have broadly found accurate matching to experimental data across brain areas providing mechanistic insights into the observed low dimensional population dynamics (Morales et al. 2023). However, the empirical success highlights a gap in theory where the neural network model used to derive the spectrum was minimal with linear neurons. In this work, we aim to close this gap by studying the covariance spectrum in nonlinear networks and broader dynamical regimes including chaos. Surprisingly, we found that the spectrum can be precisely understood by equations analogous to the linear theory while using an effective connectivity parameter reflecting the nonlinear neurons. Across the dynamical regimes, we found this effective connectivity goes through an increasing and then decreasing trend with the turning point near the edge of chaos. These results further our understanding of nonlinear neural population dynamics and provide additional theoretical support for applying the covariance spectrum in biological circuits.