  We prove a number of unconditional statistical results of the Hecke
coefficients for unitary cuspidal representations of $\operatorname{GL}(2)$
over number fields. Using partial bounds on the size of the Hecke coefficients,
instances of Langlands functoriality, and properties of Rankin-Selberg
$L$-functions, we obtain bounds on the set of places where linear combinations
of Hecke coefficients are negative. Under a mild functoriality assumption we
extend these methods to $\operatorname{GL}(n)$. As an application, we obtain a
result related to a question of Serre about the occurrence of large Hecke
eigenvalues of Maass forms. Furthermore, in the cases where the Ramanujan
conjecture is satisfied, we obtain distributional results of the Hecke
coefficients at places varying in certain congruence or Galois classes.
