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

ExtendedSL(2,R)/U(1) characters, or modular properties of a simple non-rational conformal field theory