  In this paper, we study the topological spectrum of weighted Birkhoff
averages over aperiodic and irreducible subshifts of finite type. We show that
for a uniformly continuous family of potentials, the spectrum is continuous and
concave over its domain. In case of typical weights with respect to some
ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case
of full shift and under the assumption that the potentials depend only on the
first coordinate, we show that our result is applicable for regular weights,
like M\"obius sequence.
