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UTe$_2$ is a recently discovered promising candidate for a spin-triplet superconductor. In contrast to conventional spin-singlet superconductivity, spin-triplet superconductivity possesses spin and angular momentum degrees of freedom. To detect these degrees of freedom and obtain the solid evidence of spin-triplet superconductivity in UTe$_2$, we have performed $^{125}$Te-NMR measurements, which are sensitive to the local spin susceptibility at a nuclear site. We reported a slight decrease in the Knight shift along the $b$ and $c$ axes ($K_b$ and $K_c$, respectively) below $T_c$ at a low magnetic field $H$. Although the decrease in $K_c$ vanished above 5.5 T, the decrease in $K_b$ was independent of $H$ up to 6.5 T. We further measured the $^{125}$Te-NMR spectra for $H \parallel b$ up to 14.5 T, and found that the decrease in $K_b$ in the SC state starts to be small around 7 T and almost zero at 12.5 T, although the superconductivity is confirmed up to 14.5 T by the $\chi_{\rm AC}$ measurements. This indicates that the SC spin state gradually changes with the application of $H$, and implies that the superconductivity possesses the spin degrees of freedom. The pressure dependence of the Knight shift has also been studied.
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