  Spectral invariants are quantitative measurements in symplectic topology
coming from Floer homology theory. We study their dependence on the choice of
coefficients in the context of Hamiltonian Floer homology. We discover
phenomena in this setting which hold for $\mathbb{Z}$-coefficients and fail for
all field coefficients. For example, we prove that the spectral norm, an
important metric derived from spectral invariants, is unbounded over
$\mathbb{Z}$ for complex projective spaces, while it is uniformly bounded over
all fields. This allows us to answer a symplectic version of a question of
Hingston, originally asked in the setting of the energy functional on the loop
space. We also provide applications to Hamiltonian dynamics and Hofer's
geometry.
