Classification of maximally non-self-dual modular categories of small dimension

We prove that a non-pointed maximally non-self-dual (MNSD) modular category of Frobenius-Perron (FP) dimension less than $2025$ has at most two possible types, and all these types can be realized except those of FP dimension $675$, $729$ and $1125$. We also prove that all these modular categories are group-theoretical except the modular categories of dimension $675$. Our result shows that a non-group-theoretical MNSD modular category of smallest FP dimension may be the category of FP dimension $675$, and non-pointed MNSD modular category of smallest FP dimension is the category of FP dimension $243$.