Distinguishing newforms by their Hecke eigenvalues

Let f , g be distinct normalized Hecke eigenforms of weights k 1 , k 2 lying in the subspace of newforms with Fourier coefficients { n ( k 1 - 1 ) / 2 λ f ( n ) } n ∈ N and { n ( k 2 - 1 ) / 2 λ g ( n ) } n ∈ N respectively. For such newforms f , g of CM type and primes p , we study the natural density of the set S = { p | λ f ( p ) = λ g ( p ) } . We show that the upper natural density of S is ≤ 3 / 4 if f ≠ g and it is equal to 1/2 when f and g have different weights and have the same associated CM (quadratic) field. Further, f and g have different associated CM (quadratic) fields if and only if the natural density of S is 1/4. When at least one of f , g is a non-CM form, we study the natural density of the sets S + ( x , α ) = { p ≤ x | θ f ( p ) + θ g ( p ) = α } and S - ( x , β ) = { p ≤ x | θ f ( p ) - θ g ( p ) = β } where θ f ( p ) , θ g ( p ) ∈ [ 0 , π ] are the angles associated to the p -th Hecke eigen values of f , g respectively and α ∈ [ 0 , 2 π ] , β ∈ [ - π , π ] . In this case, we show that S + ( x , α ) and S - ( x , β ) have natural density zero when f and g are distinct and not twists of each other. Finally, we establish an explicit link between the elements of these sets and sign changes of Fourier coefficients at prime powers which allows us to improve a number of existing results in this set-up.