Petersson conjectured that and are always complex conjugate,
which implies

(5)

and

(6)

This conjecture was proven by Deligne (1974), which also proved the tau conjecture as a special case. Deligne was awarded the Fields
medal for his proof.

Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag,
p. 140, 1997.Deligne, P. "La conjecture de Weil. I."
Inst. Hautes Études Sci. Publ. Math.43, 273-307, 1974.Deligne,
P. "La conjecture de Weil. II." Inst. Hautes Études Sci. Publ.
Math.52, 137-252, 1980.