  The vanishing affine connections have been used solely while adopting the
modified $f(Q)$ gravity theory to the cosmology. Consequently, researchers
could not get beyond what is already known in $f(T)$ theory earlier. To
alleviate this problem, in the present manuscript we investigate a recently
proposed construction of $f(Q)$ theory using non-vanishing affine connection in
the spatially flat FLRW spacetime. We then investigate the cosmological
solutions of $f(Q)$ theory for a perfect fluid through the phase space
analysis. We introduce few variables and dimensionless parameters to construct
the corresponding equations suitable for the dynamical system approach. The
conservation of the energy-momentum tensor leads to a constraint equation that
relates the dynamical variables. Briefly, both unstable and stable de Sitter
solutions appear which correspond to early and late times accelerated
expansions. Also, unstable points corresponding to the matter dominated and
radiation dominated eras have been found which do exist for every $f(Q)$
function. As a result, the present discussion shows that $f(Q)$ gravity endowed
by non-vanishing affine connections is capable of explaining a true sequence of
cosmic eras.
