  We compute the trivial source character tables (also called species tables of
the trivial source ring) of the infinite family of finite groups SL(2,q) over a
large enough field of positive characteristic $\ell$ via character-theoretical
methods in the cases in which $q$ is odd, $\ell \mid (q\pm1)$ when~$\ell$ is
odd, and $q\equiv \pm 3\pmod{8}$ when $\ell=2$.
