The vacuum is unique, so $N_0=1$. The stress tensor is the spin-$2$ irreducible representation of $SU(2)$ and has $2j+1=5$ states. At the next level, a translation of spin $1$ gives
$$
1\otimes2=3\oplus2\oplus1.
$$
Stress-tensor conservation makes the spin-$1$ divergence a null <conformal descendant>, leaving dimensions $7+5$. Hence
$$
\boxed{N_0=1,
\qquad N_1=5,
\qquad N_2=12.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 1 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-354/1/a/ii/solution