Let $C$ be a complex smooth projective curve of genus $g$, $C_d$ its d-fold symmetric product and $C^d$ its d-fold fiber product. Let $L$ a line bundle on $C$, and $L\boxtimes\cdots\boxtimes L$ (d copies) the coresponding line bundle on $C^d$. Since $L\boxtimes\cdots\boxtimes L$ is invatiant under the action of the symmetric group $S_d$, it descends to a line bundle $\widetilde{L}$ on $C_d$.

My question is:

Can $\widetilde{L}$ be expressed by $\theta$ and $x$, here $\theta$ is the class of the pull back of the theta divisor and $x$ is the class of $C_{d-1}$ in $C_d$?

Is $\widetilde{K_C}$ the canonical bundle of $C_d$?