On the Distribution of Frobenius of Weight 2 Eigenforms with Quadratic Coefficient Field

TL;DR

This paper introduces a heuristic model predicting the distribution of primes for which the coefficients of weight 2 eigenforms with quadratic coefficient fields are rational integers, supported by numerical data.

Contribution

It provides a new heuristic model for the asymptotic distribution of such primes in weight 2 eigenforms with quadratic fields, without inner twists.

Findings

Numerical data aligns with the proposed model.

The model accurately predicts the distribution of primes with rational coefficients.

Insights into the behavior of eigenform coefficients over quadratic fields.

Abstract

In this article we present a heuristic model that describes the asymptotic behaviour of the number of primes p such that the p-th coefficient of a given eigenform is a rational integer. We treat the case of a weight 2 eigenform with quadratic coefficient field without inner twists. Moreover we present numerical data which agrees with our model and the assumptions we made to obtain it.