  Using a set of cosmological hydro-dynamical simulations, we constrained the
properties of primordial magnetic fields by studying their impact on the
formation and evolution of dwarf galaxies. We performed a large set of
simulations (8 dark matter only and 72 chemo-hydrodynamical) including
primordial magnetic fields through the extra density fluctuations they induce
at small length scales ($k \geq 10\,h\,\rm{Mpc^{-1}}$) in the matter power
spectrum. We explored a large variety of primordial magnetic fields with
strength $B_\lambda$ ranging from $0.05$ to $0.50\,\textrm{nG}$ and magnetic
energy spectrum slopes $n_B$ from $-2.9$ to $-2.1$. Strong magnetic fields
characterized by a high amplitude ($B_\lambda=0.50,\,0.20\,\textrm{nG}$ with
$n_B=-2.9$) or by a steep initial power spectrum slope ($n_B=-2.1,-2.4$, with
$B_\lambda=0.05\,\textrm{nG}$) induce perturbations in the mass scales from
$10^7$ to $10^9\,\rm{M}_{\odot}$. In this context emerging galaxies see their
star formation rate strongly boosted. They become more luminous and metal rich
than their counterparts without primordial magnetic fields. Such strong fields
are ruled out by their inability to reproduce the observed scaling relations of
dwarf galaxies. They predict dwarf galaxies to be at the origin of an
unrealistically early reionization of the Universe and also overproduce
luminous satellites in the Local Group. Weaker magnetic fields impacting the
primordial density field at corresponding masses $\lesssim
10^6\,\rm{M}_{\odot}$, produce a large number of mini dark halos orbiting the
dwarfs, however out of reach for current lensing observations. This study
allows for the first time to constrain the properties of primordial magnetic
fields based on realistic cosmological simulations of dwarf galaxies.
