Trivial source character tables of SL(2,q)

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 $\mathrm{\ell }$ via character-theoretical methods in the cases in which $q$ is odd, $\mathrm{\ell }\mid (q\pm 1)$ when~$\mathrm{\ell }$ is odd, and $q\equiv \pm 3(\mathrm{m}\mathrm{o}\mathrm{d}8)$ when $\mathrm{\ell }=2$.