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As shown by Cardy \cite{Cardy}, modular invariance of the partition function
of a given unitary non-singular $2d$ CFT with left and right central
charges $c_L$ and $c_R$, implies that the density of states in a
microcanonical ensemble, at excitations $\Delta$ and $\bar\Delta$ and in the
{saddle point approximation}, is $\rho_0(\Delta,\bar\Delta; c_L, c_R)=c_L
\exp(2\pi\sqrt{{c_L\Delta}/{6}}\ ) \cdot c_R \exp(2\pi\sqrt{{c_R\bar\Delta}/
{6}}\ )$. In this paper, we extend Cardy's analysis and show that up to
contributions which are {\em exponentially suppressed} compared to the
leading Cardy's result, the density of states takes the form $\rho
(\Delta,\bar\Delta; c_L, c_R)= f(c_L\Delta) f(c_R\bar\Delta)\rho_0
(\Delta,\bar\Delta; c_L, c_R)$, for a function $f(x)$ which we specify. In
particular, we show that \textrm{(i)} $\rho(\Delta,\bar\Delta; c_L, c_R)$
is the product of contributions of left and right movers and hence, to this
approximation, the partition function of any modular invariant, non-singular
unitary $2d$ CFT is holomorphically factorizable, \textrm{(ii)}
$\rho (\Delta,\bar\Delta; c_L, c_R)/(c_Lc_R)$ is only a function of
$c_L\Delta$ and $c_R\bar\Delta$ and, \textrm{(iii)} treating $\rho
(\Delta,\bar\Delta; c_L, c_R)$ as the density of states of a microcanonical
ensemble, we compute the entropy of the system in the canonical counterpart
and show that the function $f(c\Delta)$ is such that the canonical entropy,
up to exponentially suppressed contributions, is simply given by the
Cardy's result $\ln \rho_0(\Delta,\bar\Delta; c_L, c_R)$.
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