Real (lower panel) and imaginary (upper panel) parts of normalized phase speed, v_ph / v_Ai = ω / (k_zv_Ai), vs. normalized wave number k_za of unstable harmonics for different values of the Alfvén Mach number (after the numerical solution of the dispersion equation Eq. (26)). Red, green, and blue lines indicate the phase speeds and the growth rates of m =  − 2 harmonics for the Alfvén Mach number M_A = 0.7, m =  − 3 harmonics for the Alfvén Mach number M_A = 0.5 and m =  − 4 harmonics for the Alfvén Mach number M_A = 0.3, respectively. Here, we assume the following parameters: ρ_i / ρ_e = 0.67, ε_i = B_iφ / B_iz = Aa / B_iz = 0.02, ε_e = B_eφ / B_ez = 0.01, B_iz / B_ez = 1.