  Recently topological aspects of magnon band structure have attracted much
interest, and especially, the Dirac magnons in Cu3TeO6 have been observed
experimentally. In this work, we calculate the magnetic exchange interactions
J's using the first-principles linear-response approach and find that these J's
are short-range and negligible for the Cu-Cu atomic pair apart by longer than 7
Angstrom. Moreover there are only 5 sizable magnetic exchange interactions, and
according to their signs and strengths, modest magnetic frustration is
expected. Based on the obtained magnetic exchange couplings, we successfully
reproduce the experimental spin-wave dispersions. The calculated neutron
scattering cross section also agrees very well with the experiments. We also
calculate Dzyaloshinskii-Moriya interactions (DMIs) and estimate the canting
angle (about 1.3{\deg}) of the magnetic non-collinearity based on the
competition between DMIs and J's, which is consistent with the experiment. The
small canting angle agrees with that the current experiments cannot distinguish
the DMI induced nodal line from a Dirac point in the spin-wave spectrum.
Finally we analytically prove that the "sum rule" conjectured in [Nat. Phys.
14, 1011 (2018)] holds but only up to the 11th nearest neighbour.
