The emission of electrons from self-assembled InAs quantum dots (SAQDs) is probed with transient capacitance spectroscopy (DLTS). The SAQDs are epitaxially grown on (100) GaAs and embedded in a slightly doped Schottky diode. Unprecedented resolution in our measurements enables us to determine different energies for emission from the s shell and the p shell of the quantum dots and to observe a strong field dependence of the activation energies derived from our DLTS spectra. Furthermore, we resolve different DLTS peaks for the emission from singly and doubly occupied s shells. The analysis of our data reveals that both the electric field as well as charge-state dependence of our spectra can be explained by a model in which phonon-assisted tunneling plays a crucial role for emission from the quantum-dot s state.