The scattering properties of plasmonic nanostructures arranged in two-dimensional periodic arrays are analyzed by expanding their response into perfectly emitting and absorbing modes. It is shown that the frequencies of these modes determine the shape of the reflection and transmission spectra in the same way that the positions of point-like charges determine the electric field around them. This helps to develop a visual interpretation of many resonant effects which occur due to overlapping resonances, and the Fano interference effect is considered as a basic example. This approach works naturally in the domain of complex frequencies and provides a systematic tool to analyze, optimize, and engineer the resonant properties of nanostructures for various applications.