  The level-k U(1) Chern-Simons theory is a spin topological quantum field
theory for k odd. Its dynamics is captured by the 2d CFT of a compact boson
with a certain radius. Recently it was recognized that a dependence on the 2d
spin structure can be given to the CFT by modifying it using the so-called Arf
invariant. We demonstrate that one can reorganize the torus partition function
of the modified CFT into a finite sum involving a finite number of conformal
blocks. This allows us to reproduce the modular matrices of the spin theory. We
use the modular matrices to calculate the partition function of the spin
Chern-Simons theory on the lens space $L(a,\pm 1)$, and demonstrate the
expected dependence on the 3d spin structure.
