Double coset operators and eta quotients

Abstract

We study a sort of generalized double coset operators, which feature changing the characters of modular forms. For any pair of characters v1 and v2, we describe explicitly those operators mapping modular forms of character v1 to those of v2. We give three applications, concerned with eta-quotients. For the first application, we give many pairs of etaquotients 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 formulae of the coefficients of these functions, generalizing Newman's theorem. For the third application, we decribe 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.