Average estimates and sign change of Fourier coefficients of cusp forms at integers represented by binary quadratic form of fixed discriminant

TL;DR

This paper studies the average behavior and sign changes of Fourier coefficients of Hecke eigenforms at integers represented by fixed discriminant binary quadratic forms, providing new insights into their distribution and oscillation patterns.

Contribution

It establishes the average behavior of Fourier coefficients supported on integers represented by primitive positive definite binary quadratic forms with fixed discriminant and class number one, and quantifies their sign changes.

Findings

Average Fourier coefficient behavior established for fixed discriminant forms.

Quantitative bounds on the number of sign changes of Fourier coefficients.

Results apply to integers represented by primitive positive definite binary quadratic forms.

Abstract

In this article, we establish an average behaviour of the normalised Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$. We also obtain a quantitative result for the number of sign changes of the sequence of the normalised Fourier coefficients $\lambda_{f}(n)$ of the Hecke eigenforms $f$ where $n$ is represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$ in the interval $(x,2x]$, for sufficiently large $x$.