  With the recently discovered formula, which relates the off shell Euclidean
one loop determinant to the on shell quantities such as normal and quasinormal
modes in real spacetime, we work out the one loop partition function for
$\mathcal{N}=1$ supergravity in AdS$_3$ from scratch. In passing, we also
provide the explicit expression for one loop determinant of a field with
arbitrary spin in AdS$_3$. To achieve this, we firstly derive the determinant
expression for the one loop partition function in question using the powerful
decomposition technique, and then we construct the normal modes in a purely
algebraic way by demonstrating that the space of normal modes falls into the
representation of SL(2,$R$) Lie algebra associated to AdS$_3$. The whole
procedure developed here turns out to be much simpler than the previous
strategy.
