  Modular data is the most significant invariant of a modular tensor category.
We pursue an approach to the classification of modular data of modular tensor
categories by building the modular $S$ and $T$ matrices directly from
irreducible representations of $SL_2(\mathbb{Z}/n \mathbb{Z})$. We discover and
collect many conditions on the $SL_2(\mathbb{Z}/n \mathbb{Z})$ representations
to identify those that correspond to some modular data. To arrive at concrete
matrices from representations, we also develop methods that allow us to select
the proper basis of the $SL_2(\mathbb{Z}/n \mathbb{Z})$ representations so that
they have the form of modular data. We apply this technique to the
classification of rank-$6$ modular tensor categories, obtaining a
classification up to modular data. Most of the calculations can be automated
using a computer algebraic system, which can be employed to classify modular
data of higher rank modular tensor categories.
