A methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained in a rotating spherical shell, and to study their stability is presented. It is based on continuation, Newton-Krylov, and Arnoldi methods. An application to the study of a double-Hopf bifurcation of the basic state is shown for Ekman and Prandtl numbers E=10−4 and σ=0.1, respectively, radius ratios η∈[0.32,0.35], Rayleigh numbers R∈[1.8×105,6×105], and nonslip and perfectly conducting boundary conditions. The knowledge of the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stable thermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of some new intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of the eigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves.