Double coset operators and eta-quotients

TL;DR

This paper investigates generalized double coset operators that transform modular forms with different characters, providing explicit descriptions, applications to eta-quotients, recursive formulas, and an algorithm for expressing eta-powers as eta-quotients.

Contribution

It explicitly characterizes operators between modular forms with different characters and applies these to eta-quotients, deriving new recursive formulas and an expression criterion.

Findings

Identified operators mapping eta-quotients of small weights and levels.

Derived recursive formulas for eta-power coefficients up to exponent 24.

Developed an algorithm to express eta-powers as linear combinations of eta-quotients.

Abstract

We study a type of generalized double coset operators which may change the characters of modular forms. For any pair of characters $v_1$ and $v_2$, we describe explicitly those operators mapping modular forms of character $v_1$ to those of $v_2$. We give three applications, concerned with eta-quotients. For the first application, we give many pairs of eta-quotients of small weights and levels, such that there are operators maps one eta-quotient to another. We also find out these operators. For the second application, we apply the operators to eta-powers whose exponents are positive integers not greater than $24$. This results in recursive formulas of the coefficients of these functions, generalizing Newman's theorem. For the third application, we describe a criterion and an algorithm of whether and how an eta-power of arbitrary integral exponent can be expressed as a linear combination of certain eta-quotients.