So I have a slight confusion about the meaning of $\Phi_{1,3}$ is problem 1 of the new homework. Is $\Phi_{1,3}$ referring to the field whose corresponding state $|\Phi_{1,3}\rangle$ satisfies $L_0|\Phi_{1,3}\rangle=h_{1,3}|\Phi_{1,3}\rangle$ and $L_n|\Phi_{1,3}\rangle=0$ for all $n$ bigger than $0$?

Correct.

$\Phi_{1,3}$ is the field and $|\Phi_{1,3}\rangle$ the corresponding state, which is a Virasoro primary with a null vector at level $1\times3=3$.

The problem is to find an explicit expression for that null vector (as a linear combination of $L_{-3}|\Phi_{1,3}\rangle$, $L_{-2}L_{-1}|\Phi_{1,3}\rangle$ and $L_{-1}^3|\Phi_{1,3}\rangle$).

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