We present measurements of the $E$-mode polarization angular auto-power spectrum ($EE$) and temperature-$E$-mode cross-power spectrum ($TE$) of the cosmic microwave background (CMB) using 150 GHz data from three seasons of SPTpol observations. We report the power spectra over the spherical harmonic multipole range $50 1050$ and $\ell > 1475$, respectively. The observations cover 500 deg$^2$, a fivefold increase in area compared to previous SPTpol analyses, which increases our sensitivity to the photon diffusion damping tail of the CMB power spectra enabling tighter constraints on \LCDM model extensions. After masking all sources with unpolarized flux $>50$ mJy we place a 95% confidence upper limit on residual polarized point-source power of $D_\ell = \ell(\ell+1)C_\ell/2π 1000$ results in a preference for a higher value of the expansion rate ($H_0 = 71.3 \pm 2.1\,\mbox{km}\,s^{-1}\mbox{Mpc}^{-1}$ ) and a lower value for present-day density fluctuations ($σ_8 = 0.77 \pm 0.02$).
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