  We analyse the $T\bar{T}$ deformation of 2d CFTs in a special double-scaling
limit, of large central charge and small deformation parameter. In particular,
we derive closed formulae for the deformation of the product of left and right
moving CFT characters on the torus. It is shown that the $1/c$ contribution
takes the same form as that of a CFT, but with rescalings of the modular
parameter reflecting a state-dependent change of coordinates. We also extend
the analysis for more general deformations that involve $T\bar{T}$, $J\bar{T}$
and $T\bar{J}$ simultaneously. We comment on the implications of our results
for holographic proposals of irrelevant deformations.
