Representations of $SL_2(\mathbb{F}_3)$

Question

I am trying to determine all the irreducible representations of the group $SL_2(\mathbb{F}_3)$.I have determined its character table and I have seen that there is a unique $2$-dimensional representation (say $V$) of the group which has real character and other $2$-dimensional representation can be expressed in terms of it like the other two $2$-dimensional representations are simply the tensor product of $V$ with the two $1$-dimensional representations whose characters are complex conjugates.Also the unique $3$-dimensional representation occurs as the direct summand with the trivial one in the decomposition of the representation obtained by tensoring $V$ twice.So everything hinges on finding the representation $V$.I am really struggling with it. Any help is appreciated.