  We revisit the flavored elliptic genus of the N=2 superconformal cigar model
and generalize the analysis of the path integral result to the case of real
central charge. It gives rise to a non-holomorphic modular covariant function
generalizing completed mock modular forms. We also compute the genus for
angular orbifolds of the cigar and Liouville theory and decompose it in terms
of discrete and continuous contributions. The orbifolded elliptic genus at
fractional level is a completed mock modular form with a shadow related to
U$(1)$ modular invariants at rational radius squared. We take the limit of the
orbifolded genera towards a weighted ground state index and carefully interpret
the contributions. We stress that the orbifold cigar and Liouville theories
have a maximal and a minimal radius, respectively.
