Variants of Lehmer's speculation for newforms (2005.10354v4)

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 is a value of $\tau(n)$ 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 $3\leq \ell\leq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $\tau(n)$ with $n>1$, we prove that $$\tau(n)\not \in {\pm 1, \pm 3, \pm 5, \pm 7, \pm 13, \pm 17, -19, \pm 23, \pm 37, \pm 691},$$ and assuming GRH we show for primes $\ell$ that $$\tau(n)\not \in \left { \pm \ell\ : \ 41\leq \ell\leq 97 \ {\textrm{with}}\ \left(\frac{\ell}{5}\right)=-1\right} \cup \left { -11, -29, -31, -41, -59, -61, -71, -79, -89\right}. $$ 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 $\ell$, we prove that $\pm \ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{\pm}(\ell,m)=O_{\ell}(m)$ and even level coprime to $\ell,$ where $M^{\pm}(\ell,m)$ is effectively computable.