  Let r : G_Q -> GL_n Q_l be a motivic l-adic Galois representation. For fixed
m > 1 we initiate an investigation of the density of the set of primes p such
that the trace of the image of an arithmetic Frobenius at p under r is an m^th
power residue modulo p. Based on numerical investigations with modular forms we
conjecture (with Ramakrishna) that this density equals 1/m whenever the image
of r is open. We further conjecture that for such r the set of these primes p
is independent of any set defined by Cebatorev-style Galois theoretic
conditions (in an appropriate sense). We then compute these densities for
certain m in the complementary case of modular forms of CM-type with rational
Fourier coefficients; our proofs are a combination of the Cebatorev density
theorem (which does apply in the CM case) and reciprocity laws applied to Hecke
characters. We also discuss a potential application (suggested by Ramakrishna)
to computing inertial degrees at p in abelian extensions of imaginary quadratic
fields unramified away from p.
