We define extended SL(2,R)/U(1) characters which include a sum over winding
sectors. By embedding these characters into similarly extended characters of
N=2 algebras, we show that they have nice modular transformation properties. We
calculate the modular matrices of this simple but non-trivial non-rational
conformal field theory explicitly . As a result, we show that discrete SL(2,R)
representations mix with continuous SL(2,R) representations under modular
transformations in the coset conformal field theory. We comment upon the
significance of our results for a general theory of non-rational conformal
field theories.Comment: JHEP style, 25 pages, 2 figures, v2: minor corrections, reference
added, version to appear in JHE
