  Using 586 pb$^{-1}$ of $e^+e^-$ annihilation data taken with the CLEO-c
detector at $\sqrt{s}=4170$ MeV, the peak of the charmonium resonance
$\psi(4160)$, we analyze the decay $\psi(4160)\to \pi^+\pi^- J/\psi$, and
report observation of the charged state $Z_c^\pm(3900)$ decaying into $\pi^\pm
J/\psi$ at a significance level of $>5 \sigma$. We obtain
$M(Z_c^\pm)=3886\pm4(\text{stat})\pm 2(\text{syst})$ MeV and
$\Gamma(Z_c^\pm)=37\pm4(\text{stat})\pm 8(\text{syst})$ MeV, which are in good
agreement with the results for this resonance obtained by the BES III and Belle
Collaborations in the decay of the resonance Y(4260). We also present first
evidence for the production of the neutral member of this isospin triplet,
$Z_c^0(3900)$ decaying into $\pi^0J/\psi$ at a $3.5\sigma$ significance level.
