  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.
