Using 16 energy points of $e^{+}e^{-}$ annihilation data collected in the vicinity of the $J/psi$ resonance with the BESIII detector and with a total integrated luminosity of around 100 pb$^{-1}$, we study the relative phase between the strong and electromagnetic amplitudes of $J/psi$ decays. The relative phase between $Jpsi$ electromagnetic decay and the continuum process ($e^{+}e^{-}$ annihilation without the $J/psi$ resonance) is confirmed to be zero by studying the cross section lineshape of $mu^{+}mu^{-}$ production. The relative phase between $J/psi$ strong and electromagnetic decays is then measured to be $(84.9pm3.6)^circ$ or $(-84.7pm3.1)^circ$ for the $2(pi^{+}pi^{-})pi^{0}$ final state by investigating the interference pattern between the $J/psi$ decay and the continuum process. This is the first measurement of the relative phase between $J/psi$ strong and electromagnetic decays into a multihadron final state using the lineshape of the production cross section. We also study the production lineshape of the multihadron final state $etapi^{+}pi^{-}$ with $etatopi^{+}pi^{-}pi^{0}$, which provides additional information about the phase between the $J/psi$ electromagnetic decay amplitude and the continuum process. Additionally, the branching fraction of $J/psito 2(pi^{+}pi^{-})pi^{0}$ is measured to be $(4.73pm0.44)%$ or $(4.85pm0.45)%$, and the branching fraction of $J/psitoetapi^{+}pi^{-}$ is measured to be $(3.78pm0.68)times10^{-4}$. Both of them are consistent with the world average values. The quoted uncertainties include both statistical and systematic uncertainties, which are mainly caused by the low statistics.
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