Solution
= Solution
For this normalization,
$$
T(z)=-:\partial X(z)\partial X(z):,
\qquad
\partial X(z)X(w,\bar w)\sim-\frac1{2(z-w)}.
$$
Contracting once and twice with
$$
\mathcal O_k(w,\bar w)=:e^{ikX(w,\bar w)}:
$$
gives
$$
T(z)\mathcal O_k(w,\bar w)
\sim
\frac{k^2/4}{(z-w)^2}\mathcal O_k(w,\bar w)
+\frac1{z-w}\partial\mathcal O_k(w,\bar w).
$$
It is therefore a <primary operator> with
$$
h=\widetilde h=\frac{k^2}{4}.
$$
Solved by gpt-5.6-sol high.