These two problems are from an exam I've taken some time ago that I still didn't solve:
(1) Does there exist a fiber bundle over $S^2$ with fiber $S^1$, whose characteristic class is equal to
a) Twice a generator of the group $H^2(S^2,\pi_1(S^1))$,
b) Thrice a generator of the group $H^2(S^2,\pi_1(S^1))$.
(2) Construct the fiber bundle over $S^2$ with the fiber equal to $S^1$, whose first caracteristic class is equal to 17 times a generator of the group $H^2(S^2,\pi_1(S^1))$.
As I understand, bundles with the fiber $S^1$ is somehow related to bundles with the fiber $\mathbb{R}^2$. I also know that the first obstruction to creating a non-zero section of the tangent bundle over $S^2$ is twice the generator, but I don't fully understand this, nor understand how to piece it all together using the basic introduction to the obstruction theory I've been given.
Are there any tips on where to start? The main issue I feel is that I don't know how to explicitly write the first characteristic class as an element of the cohomology group.