Some remarks related to Maeda's conjecture

TL;DR

This paper investigates the problem of counting pairs of normalized eigenforms of a given weight and level that share the same p-th Fourier coefficient, contributing to the understanding of eigenform relationships.

Contribution

It provides new insights into the enumeration of eigenform pairs with matching Fourier coefficients at a fixed prime, relating to Maeda's conjecture.

Findings

Derived formulas for counting such eigenform pairs

Identified conditions under which eigenforms share Fourier coefficients

Enhanced understanding of eigenform distribution and symmetry

Abstract

In this article we deal with the problem of counting the number of pairs of normalized eigenforms $ (f,g) $ of weight $k$ and level $N$ such that $ a_p (f) = a_p (g) $ where $a_p (f) $ denotes the $p-$th Fourier coefficient of $f$. Here $p$ is a fixed prime.