Papers › Variants of Lehmer's speculation for newforms
Variants of Lehmer's speculation for newforms
Jennifer S. Balakrishnan, William Craig, Ken Ono, Wei-Lun Tsai
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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 is a value of τ(n) or is a Fourier coefficient a_f(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 integers. We determine infinitely many spaces for which the primes 3≤ℓ≤37 are not absolute values of coefficients of newforms with integer coefficients. For τ(n) with n>1, we prove that τ(n)∉{±1, ±3, ±5, ±7, ±13, ±17, -19, ±23, ±37, ±691}, and assuming GRH we show for primes ℓ that τ(n)∉{ ±ℓ : 41≤ℓ≤97 with (ℓ/5)=-1} ∪{ -11, -29, -31, -41, -59, -61, -71, -79, -89}. 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)=O_ℓ(m) and even level coprime to ℓ, where M^±(ℓ,m) is effectively computable.
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