We investigate the coupling between the magnetic and superconducting order parameters in an 8-m-long meander line ("wire") made of a La_1.94Sr_0.06CuO₄ film with a cross section of 0.5×100 μm2. The magnetic order parameter is determined using the low-energy muon spin relaxation technique. The superconducting order parameter is characterized by transport measurements and modified by high current density. We find that when the superconducting order parameter is suppressed by the current, the magnetic transition temperature, T_m, increases. The extracted sign and magnitude of the Ginzburg-Landau coupling constant indicate that the two orders are repulsive, and that our system is located close to the border between first- and second-order phase transition.