One loop partition function from normal modes for $ \mathcal{N}=1$ supergravity in AdS$_3$

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.