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We present the first numerical relativity waveforms for binary black hole mergers produced using spectral methods that show both the displacement and the spin memory effects. Explicitly, we use the SXS Collaboration's $\texttt{SpEC}$ code to run a Cauchy evolution of a binary black hole merger and then extract the gravitational wave strain using $\texttt{SpECTRE}$'s version of a Cauchy-characteristic extraction. We find that we can accurately resolve the strain's traditional $m=0$ memory modes and some of the $m\not=0$ oscillatory memory modes that have previously only been theorized. We also perform a separate calculation of the memory using equations for the Bondi-Metzner-Sachs charges as well as the energy and angular momentum fluxes at asymptotic infinity. Our new calculation uses only the gravitational wave strain and two of the Weyl scalars at infinity. Also, this computation shows that the memory modes can be understood as a combination of a memory signal throughout the binary's inspiral and merger phases, and a quasinormal mode signal near the ringdown phase. Additionally, we find that the magnetic memory, up to numerical error, is indeed zero as previously conjectured. Lastly, we find that signal-to-noise ratios of memory for LIGO, the Einstein Telescope (ET), and the Laser Interferometer Space Antenna (LISA) with these new waveforms and new memory calculation are larger than previous expectations based on post-Newtonian or Minimal Waveform models.