Jeff Adams has comprehensive notes on his website: http://www.math.umd.edu/~jda/characters/characters.pdf
The irreducible characters of the groups SL(2), PGL(2), GL(2) and PSL(2) over finite fields are described.
The problem for PSL(2) was originally solved by Frobenius in his 1896 paper: Über Gruppencharaktere. It is the first article in Volume 3 of his collected works (Gesammelte AbhandunglesnAbhandunglen, Band III). On the last page of this article, I found the table: