Variants of Lehmer's speculation for newforms

Abstract

In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer α is a value of τ(n) or is a Fourier coefficient af(n) of any given newform f(z). We offer a method, which applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd α. We determine infinitely many spaces for which the primes 3≤ℓ≤37 are not absolute values of coefficients of newforms with integer coefficients, and we obtain many explicit examples for τ(n). We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes ℓ, we prove that ±ℓm is not a coefficient of any such newform f with weight 2k>M±(ℓ,m) and even level coprime to ℓ, where M±(ℓ,m) are effectively computable constants that are Oℓ(m).