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Let $p<q$ be odd primes, $\rho_1$ and $\rho_2$ be irreducible representationsof $\text{SL}(2,\mathbb{Z}_p)$ and $\text{SL}(2,\mathbb{Z}_q)$ of dimensions$\frac{p+1}{2}$ and $\frac{q+1}{2}$, respectively. We show that if$\rho_1\oplus\rho_2$ can be realized as modular representation associated to amodular fusion category $\mathcal{C}$, then $q-p=4$. Moreover, if $\mathcal{C}$contains a non-trivial \'{e}tale algebra, then$\mathcal{C}\boxtimes\mathcal{C}(\mathbb{Z}_p,\eta)\cong\mathcal{Z}(\mathcal{A})$as braided fusion category, where $\mathcal{A}$ is a near-group fusion categoryof type $(\mathbb{Z}_p,p)$. And we show that there exists a non-trivial$\mathbb{Z}_2$-extension of $\mathcal{A}$ that contains simple objects ofFrobenius-Perron dimension $\frac{\sqrt{p}+\sqrt{q}}{2}$.

On the realization of a class of $\text{SL}(2,\mathbb{Z})$-representations