Distinguishing Hecke eigenforms

TL;DR

This paper refines bounds on the number of initial Fourier coefficients shared by distinct Hecke eigenforms, providing explicit, asymptotic, and numerical insights into their differences.

Contribution

It offers an explicit upper bound on shared Fourier coefficients of different weight eigenforms and improves existing asymptotic bounds with numerical validation.

Findings

Explicit upper bound on shared Fourier coefficients

Improved asymptotic bounds under certain conditions

Numerical experiments confirming bound sharpness

Abstract

We revisit a theorem of Ram Murty about the number of initial Fourier coefficients that two cuspidal eigenforms of different weights can have in common. We prove an explicit upper bound on this number, and give better conditional and unconditional asymptotic upper bounds. Finally, we describe a numerical experiment testing the sharpness of the upper bound in the case of forms of level one.