  The spectral form factor is a dynamical probe for level statistics of quantum
systems. The early-time behaviour is commonly interpreted as a characterization
of two-point correlations at large separation. We argue that this
interpretation can be too restrictive by indicating that the self-correlation
imposes a constraint on the spectral form factor integrated over time. More
generally, we indicate that each expansion coefficient of the two-point
correlation function imposes a constraint on the properly weighted
time-integrated spectral form factor. We discuss how these constraints can
affect the interpretation of the spectral form factor as a probe for
ergodicity. We propose a new probe, which eliminates the effect of the
constraint imposed by the self-correlation. The use of this probe is
demonstrated for a model of randomly incomplete spectra and a Floquet model
supporting many-body localization.
