  We study period integrals involving Whittaker functions associated to generic
irreducible Casselman-Wallach representations of $\mathrm{GL}_n(F)$, where $F$
is an archimedean local field. Via the archimedean theory of newforms for
$\mathrm{GL}_n$ developed by the first author, we prove that newforms are weak
test vectors for several period integrals, including the $\mathrm{GL}_n \times
\mathrm{GL}_n$ Rankin-Selberg integral, the Flicker integral, and the
Bump-Friedberg integral. By taking special values of these period integrals, we
deduce that newforms are weak test vectors for Rankin-Selberg periods,
Flicker-Rallis periods, and Friedberg-Jacquet periods. These results parallel
analogous results in the nonarchimedean setting proven by the second author,
which use the nonarchimedean theory of newforms for $\mathrm{GL}_n$ developed
by Jacquet, Piatetski-Shapiro, and Shalika. By combining these archimedean and
nonarchimedean results, we prove the existence of weak test vectors for certain
global period integrals of automorphic forms.
