  The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG)
can be decomposed into a TBG-like flat band Hamiltonian and a high-velocity
Dirac fermion Hamiltonian. We use Hartree-Fock mean field approach to study the
projected Coulomb interacting Hamiltonian of TSTG developed in C\u{a}lug\u{a}ru
et al. [Phys. Rev. B 103, 195411 (2021)] at integer fillings $\nu=-3, -2, -1$
and $0$ measured from charge neutrality. We study the phase diagram with
$w_0/w_1$, the ratio of $AA$ and $AB$ interlayer hoppings, and the displacement
field, which introduces an interlayer potential $U$ and hybridizes the TBG-like
bands with the Dirac bands. At small $U$, we find the ground states at all
fillings $\nu$ are in the same phases as the tensor products of a Dirac
semimetal with the filling $\nu$ TBG insulator ground states, which are
spin-valley polarized at $\nu=-3$, and fully (partially) intervalley coherent
at $\nu=-2,0$ ($\nu=-1$) in the flat bands. An exception is $\nu=-3$ with
$w_0/w_1 \gtrsim 0.7$, which possibly become a metal with competing orders at
small $U$ due to charge transfers between the Dirac and flat bands. At strong
$U$ where the bandwidths exceed interactions, all the fillings $\nu$ enter a
metal phase with small or zero valley polarization and intervalley coherence.
Lastly, at intermediate $U$, semimetal or insulator phases with zero
intervalley coherence may arise for $\nu=-2,-1,0$. Our results provide a simple
picture for the electron interactions in TSTG systems, and reveal the
connection between the TSTG and TBG ground states.
