Binary Quadratic forms and the Fourier coefficients of certain weight 1 eta-quotients

TL;DR

This paper establishes a general identity linking weight 1 eta-products to binary quadratic forms, enabling explicit Fourier coefficient formulas for specific eta-quotients across various levels.

Contribution

It introduces a new identity connecting eta-products with binary quadratic forms and derives explicit Fourier coefficient formulas for certain eta-quotients.

Findings

Derived explicit Fourier coefficient formulas for eta-quotients at levels 47, 71, 135, 648, 1024, and 1872.

Established a general identity representing eta-products of weight 1 via binary quadratic forms.

Demonstrated utility of quadratic forms in multiplicative completion of eta-quotients.

Abstract

We state and prove an identity which represents the most general eta-products of weight 1 by binary quadratic forms. We discuss the utility of binary quadratic forms in finding a multiplicative completion for certain eta-quotients. We then derive explicit formulas for the Fourier coefficients of certain eta-quotients of weight 1 and level 47, 71, 135,648 1024, and 1872.