  We introduce a new invariant, the conductor exponent, of a generic
irreducible Casselman-Wallach representation of $\mathrm{GL}_n$ that quantifies
the extent to which this representation may be ramified. We also determine a
distinguished vector, the newform, occurring with multiplicity one in this
representation, with the complexity of this vector measured in a natural way by
the conductor exponent. Finally, we show that the newform is a test vector for
$\mathrm{GL}_n \times \mathrm{GL}_n$ and $\mathrm{GL}_n \times \mathrm{GL}_{n -
1}$ Rankin-Selberg integrals when the second representation is unramified. This
theory parallels an analogous nonarchimedean theory due to Jacquet,
Piatetski-Shapiro, and Shalika; combined, this completes a global theory of
newforms for automorphic representations of $\mathrm{GL}_n$ over number fields.
By-products of the proofs include new proofs of Stade's formulae and a new
resolution of the test vector problem for archimedean Godement-Jacquet zeta
integrals.
