Papers › Fourier Coefficients of Level 1 Hecke Eigenforms
Fourier Coefficients of Level 1 Hecke Eigenforms
Mitsuki Hanada, Rachana Madhukara
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Lehmer's 1947 conjecture on whether τ(n) 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 τ(n). For example, among the odd numbers α such that |α|<99, we determine that τ(n) ∉{-9, ±15, ±21, -25, -27, -33, ±35, ±45, ±49, -55, ±63, ±77, -81, ±91 }. Moreover, under GRH, we have that τ(n) ≠-|α| and that τ(n) ∉{9,25,27,39, 75,81}. We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for ΔE₄ = ∑_(n = 1)^∞ τ₁₆(n)qⁿ, we show that τ₁₆(n) ∉ {±ℓ: 1≤ℓ≤99, ℓ is odd, ℓ≠33,55,59,67,73,83,89,91} ∪{-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.
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