  We construct the nonlinear $W(sl(N+3),sl(3))$ algebras and find the spectrum
of values of the central charge that gives rise, by contracting the
$W(sl(N+3),sl(3))$ algebras, to a $W_3$ algebra belonging to the coset
$W((sl(N+3),sl(3))/(u(1)\oplus sl(N))$. Part of the spectrum was conjectured
before, but part of it is given here for the first time. Using the tool of
embedding the $W(sl(N+3),sl(3))$ algebras into linearizing algebras, we
construct new realizations of $W_3$ modulo null fields. The possibility to
predict, within the conformal linearization framework, the central charge
spectrum for minimal models of the nonlinear $W(sl(N+3),sl(3))$ algebras is
discussed at the end.
