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

ABSTRACT

In this article, we present a heuristic model that describes the asymptotic behavior of the number of primes p such that the pth 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 agree with our model and the assumptions we made to obtain it.

KEYWORDS:

• number theory
• modular forms
• Heuristic model

2000 AMS SUBJECT CLASSIFICATION:

• 11F11
• 11-02

Acknowledgments

I wish to express my sincere gratitude to Jan Tuitman and Gabor Wiese for suggesting to me this problem, for our discussions, for their enthusiasm and for their guidance. I would also like to thank Andrew Sutherland and the anonymous referee for useful comments.