The three-dimensional flow past a fixed sphere placed within a uniform stream is investigated. This paper focuses on the second bifurcation, which is responsible for the onset of the unsteadiness. Using the highly efficient Nek5000 parallel solver together with a recently developed numerical algorithm to stabilize and accelerate the numerical solution, it was possible to identify the three-dimensional eigenmode responsible for the second bifurcation. The characteristics of this eigenmode are analyzed in detail. The value of the critical Reynolds number ReIIcr=271.8, as well as the Strouhal number of the arising limit cycle, agree well with previous experimental and numerical investigations. To further assess the nature of the instability, an adjoint-based sensitivity analysis is carried out. The structure of the direct and adjoint modes are discussed, and then the core of the instability is localized. Finally, the sensitivity of the instability to a generic base flow modification is investigated.