Let $p<q$ be odd primes, $\rho_1$ and $\rho_2$ be irreducible representations of $\text{ SL}(2,\mathbb{ Z}_p)$ 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 a modular 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 category of 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 of Frobenius-Perron dimension $\frac{ \sqrt{ p}+\sqrt{ q}}{ 2}$.
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