  We study 3d mirror symmetry with loop operators, Wilson loop and Vortex loop,
and multi-flavor mirror symmetry through utilizing the $\mathbb{RP}^2 \times
\mathbb{S}^1$ index formula. The key identity which makes the above description
work well is the mod 2 version of the Fourier analysis, and we study such
structure, the S-operation in the context of a SL$(2,\mathbb{Z})$ action on 3d
SCFTs. We observed that two types of the parity conditions basically associated
with gauge symmetries which we call $\mathcal{P}$-type and $\mathcal{CP}$-type
are interchanged under mirror symmetry. We will also comment on the
T-operation.
