$\phi , \psi \in V^*$ and $u,a \in V$ and $a=(2,-1,3)^T$. I used $\{\varepsilon_1,\varepsilon_2,\varepsilon_3\}$ as canonical basis for $V$ and $\{\varepsilon^1,\varepsilon^2,\varepsilon^3\}$ as dual canonical basis. Then I solve \begin{eqnarray*} T^{ij}_k=T(\varepsilon^i,\varepsilon^j,\varepsilon_k)=\varepsilon^i(a)\varepsilon^j(\varepsilon_k)+\varepsilon^j(a)\varepsilon^i(\varepsilon_k)=a^i\delta^j_k+a^j\delta^i_k \end{eqnarray*} Now solving for tensor contractions \begin{eqnarray*} (C_{11}T)^i=T^{ji}_j=a^j\delta^i_j+a^i\delta^j_j=a^i+a^i\text{dim}V=4a^i\\ (C_{21}T)^i=T^{ij}_j=a^i\delta^j_j+a^j\delta^i_j=a^i\text{dim}V+a^i=4a^i \end{eqnarray*}
I am stuck at this point. I don't really know how to move from this to the coordinates of the contractions. I don't know how to use components of $a$ here.
Since it is a type (2,1) tensor, it's contraction must always be (1,0) tensor. Does this mean that both contractions have coordinates $(8,-4,12)$?