  We study the fundamental self-similar solution to the SU(2) sigma model,
found by Shatah and Turok-Spergel. We give a rigorous proof for its mode
stability. Based on earlier results by the second author, the present paper
constitutes the last building block for a completely rigorous proof of the
nonlinear stability of the Shatah-Turok-Spergel wave map.
