Nonvanishing of second Hecke coefficients on Atkin–Lehner sign-pattern spaces

Prove that, for every integer m and level N coprime to m, every sign pattern σ for N, and every even weight k8 for which the relevant space has dimension at least two, the second Hecke coefficient c₂(m,N,k,σ) over S_k^σ(N) and the second Hecke coefficient c₂^{new}(m,N,k,σ) over S_k^{new,σ}(N) are nonzero.

Background

The paper studies coefficients of normalized Hecke polynomials on Atkin–Lehner eigenspaces S_kσ(N) and their new subspaces S_k{new,σ}(N). Its asymptotic results establish eventual sign behavior for several families of coefficients, but the second coefficient can exhibit opposing signs across infinite families of levels and sign patterns. Nonvanishing is therefore a distinct issue from asymptotic sign determination.

The authors explicitly formulate a conjecture asserting nonvanishing of the second Hecke coefficient for all even weights k≥8, subject to the natural dimension condition dim S_kσ(N)≥2 (and its analogous condition for the newspace). They report computational verification for 8≤k≤40, 1≤m≤50, 1≤N≤300, and all sign patterns in that range, but leave the general claim unresolved.

References

Finally, we note an open problem related to the results of this paper. We conjecture that the second Hecke coefficients are always non-vanishing over $S_k\sigma(N)$ and $S_k{,\sigma}(N)$ for weight $k \ge 8$.

— Asymptotics of Hecke polynomial coefficients on the Atkin-Lehner eigenspaces  (2608.18497 - Nelson et al., 19 Aug 2026) in Section 1, immediately before Conjecture 1 (labelled \ref{conj:nonvanishing})