Local systems with tame, and a unipotent, local monodromy

We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two (Q) over bar (l)-smooth sheaves on X-1-S-1 , where X-1 is a smooth projective absolutely irreducible curve of genus g over a finite field F-q and S-1 is a reduced divisor, with pre-specified tamely ramified monodromy data at S, including precisely one point of principal unipotent monodromy, twisted by a tame character. Equivalently, we compute the number of the corresponding automorphic representations. The approach is based on using an explicit form of the trace formula for GL(2) , extending the work Counting local systems with tame ramification to include a Steinberg (= special) component, twisted by a tame character, by employing a pseudo-coefficient thereof.