Product formulas for weight two newforms

TL;DR

This paper explores product formulas for weight two newforms associated with elliptic curves over rationals, revealing that for certain curves, the coefficients form an increasing sequence of positive integers.

Contribution

It introduces a specific product expansion for weight two newforms and identifies cases where the sequence of exponents is increasing and positive.

Findings

For some elliptic curves, the sequence g_n is increasing.

The coefficients g_n are integers.

The product formula relates to the structure of the associated newform.

Abstract

For a weight two newform $f$ attached to an elliptic curve $E$ defined over rational numbers we write $f=q\prod_{n=1}^\infty (1-q^n)^{g_n}, \ g_n\in\Z$ and we observe that for some special elliptic curves $g_n$ is an increasing sequence of positive integers.