Fourier coefficients of level 1 Hecke eigenforms
Volume 200 / 2021
Abstract
Lehmer’s 1947 conjecture that $\tau (n)$ never vanishes is still unresolved. In this context, it is natural to consider variants of Lehmer’s conjecture. We determine many integers that cannot be values of $\tau (n)$. For example, among the odd numbers $\alpha $ such that $|\alpha | \leq 99$, we determine that $$ \tau (n) \notin \{-9, \pm 15, \pm 21, -25, -27, -33, \pm 35, \pm 45, \pm 49, -55, \pm 63, \pm 77, -81, \pm 91 \}.$$ Moreover, under GRH, we have $\tau (n) \neq -|\alpha |$ and $\tau (n) \notin \{9,25,27,39,75, 81\}.$ We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for $\Delta E_4 = \sum _{n = 1}^{\infty } \tau _{16}(n)q^n$, we show that $$ \tau _{16}(n) \notin \{\pm \ell : 1\leq \ell \leq 99,\, \ell \text { is odd},\, \ell \neq 33,55,59,67,73,83,89,91\} \cup \{-33,-55,-59,-67,-89,-91\}.$$ Furthermore, we implement congruences given by Swinnerton-Dyer to rule out additional large primes which divide numerators of specific Bernoulli numbers. To obtain these results, we make use of the theory of Lucas sequences, methods for solving high degree Thue equations, Barros’ algorithm for solving hyperelliptic equations, and the theory of continued fractions.
