  We show that peculiar velocities of Type Ia supernovae can be used to derive
constraints on the sum of neutrino masses, $\Sigma m_{\nu}$, and dark energy
equation of state, $w = w_0+w_a(1-a)$, from measurements of the
magnitude-redshift relation, complementary to galaxy redshift and weak lensing
surveys. Light from a supernova propagates through a perturbed Universe so the
luminosity distance is modified from its homogeneous prediction. This
modification is proportional to the matter density fluctuation and its time
derivative due to gravitational lensing and peculiar velocity respectively. At
low redshifts, the peculiar velocity signal dominates while at high redshifts
lensing does. We show that using lensing and peculiar velocity of supernovae
from the upcoming surveys WFIRST and ZTF, without other observations, we can
constrain $\Sigma m_{\nu} \lesssim 0.31$ eV, $\sigma(w_0) \lesssim 0.02$, and
${\sigma(w_a)} \lesssim 0.18$ ($1-\sigma$ CL) in the $\Sigma
m_{\nu}$-$w_0$-$w_a$ parameter space, where all the other cosmological
parameters are fixed. We find that adding peculiar velocity information from
low redshifts shrinks the volume of the parameter ellipsoid in this space by
$\sim 33$%. We also allow $\Omega_{\text{CDM}}$ to vary as well as $\Sigma
m_{\nu}$, $w_0$ and $w_a$, and demonstrate how these constraints degrade as a
consequence.
