What question did this study set out to answer?

To investigate the growth behavior of ratios of Hecke eigenvalues for various newforms.

January 18, 2026

Growth of Hecke eigenvalue ratios in newforms

Key Points

To investigate the growth behavior of ratios of Hecke eigenvalues for various newforms.
Analyzed nonzero Hecke eigenforms under specific conditions.
Developed mathematical proofs regarding eigenvalue behaviors over primes.
Examined relationships between distinct Hecke eigenvalues.
Proven that non-constant Hecke eigenforms exhibit distinct growth in eigenvalue ratios.
Identified specific conditions that affect the ratio growth of eigenvalues.

Abstract

Let Formula: see text be nonzero Hecke eigenforms on Formula: see text. For Formula: see text, let Formula: see text be the number of ratios of Formula: see text for primes Formula: see text, where Formula: see text denotes the Hecke eigenvalue of Formula: see text at Formula: see text. In this paper, we prove that if Formula: see text is not a constant multiple of Formula: see text for distinct Formula: see text, then Formula: see text

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