  Using exact diagonalization, we study the projected Hamiltonian with Coulomb
interaction in the 8 flat bands of first magic angle twisted bilayer graphene.
Employing the U(4) (U(4)$\times$U(4)) symmetries in the nonchiral (chiral) flat
band limit, we reduced the Hilbert space to an extent which allows for study
around $\nu=\pm 3,\pm2,\pm1$ fillings. In the first chiral limit $w_0/w_1=0$
where $w_0$ ($w_1$) is the $AA$ ($AB$) stacking hopping, we find that the
ground-states at these fillings are extremely well-described by Slater
determinants in a so-called Chern basis, and the exactly solvable charge $\pm1$
excitations found in [arXiv:2009.14200] are the lowest charge excitations up to
system sizes $8\times8$ (for restricted Hilbert space) in the chiral-flat
limit. We also find that the Flat Metric Condition (FMC) used in
[arXiv:2009.11301,2009.11872,2009.12376,2009.13530,2009.14200] for obtaining a
series of exact ground-states and excitations holds in a large parameter space.
For $\nu=-3$, the ground state is the spin and valley polarized Chern insulator
with $\nu_C=\pm1$ at $w_0/w_1\lesssim0.9$ (0.3) with (without) FMC. At
$\nu=-2$, we can only numerically access the valley polarized sector, and we
find a spin ferromagnetic phase when $w_0/w_1\gtrsim0.5t$ where $t\in[0,1]$ is
the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase
otherwise, confirming the perturbative calculation [arXiv:2009.13530]. The
analytic FMC ground state is, however, predicted in the intervalley coherent
sector which we cannot access [arXiv:2009.13530]. For $\nu=-3$ with/without
FMC, when $w_0/w_1$ is large, the finite-size gap $\Delta$ to the neutral
excitations vanishes, leading to phase transitions. Further analysis of the
ground state momentum sectors at $\nu=-3$ suggests a competition among
(nematic) metal, momentum $M_M$ ($\pi$) stripe and $K_M$-CDW orders at large
$w_0/w_1$.
