  We analyze the modular properties of the effective CFT description for paired
states, proposed in cond-mat/0003453, corresponding to the non-standard filling
nu =1/(p+1). We construct its characters for the twisted and the untwisted
sector and the diagonal partition function. We show that the degrees of freedom
entering our partition function naturally go to complete a Z_2-orbifold
construction of the CFT for the Halperin state. Different behaviours for the p
even and p odd cases are also studied. Finally it is shown that the tunneling
phenomenon selects out a twist invariant CFT which is identified with the
Moore-Read model.
