The $RO(\mathrm{\Pi }B)$-graded $C_2$-equivariant ordinary cohomology of $B_{C_2}U(1)$

We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $C{P}_{C_2}^{\mathrm{\infty }}=B_{C_2}U(1)$, the complex projective space, a model for the classifying space for $C_2$-equivariant complex line bundles. The $RO(C_2)$-graded Bredon ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators. These generators include the Euler class of the tautological bundle, which lies outside of the $RO(C_2)$-graded theory.