Elliptic curves and spin

In the early 2000s, Ramakrishna asked the question: For the elliptic curve E: y² = x³ - x, what is the density of primes p for which the Fourier coefficient aₚ (E) is a cube modulo p? As a generalization of this question, Weston--Zaurova formulated conjectures concerning the distribution of power residues of degree m of the Fourier coefficients of elliptic curves E/Q with complex multiplication. In this paper, we prove their conjecture for cubic residues using the analytic theory of spin. Our proof works for all elliptic curves E with complex multiplication.