Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts

Given an elliptic cusp form f and an automorphic form f ′ on a definite quaternion algebra over ℚ, there is a theta lifting from ( f, f ′) to an automorphic form ℒ ( f, f ′) on the quaternion unitary group GSp (1, 1) generating quaternionic discrete series at the Archimedean place. The aim of this paper is to provide an explicit formula for Fourier coefficients of ℒ ( f, f ′) in terms of periods of f and f ′ with respect to a unitary character χ of an imaginary quadratic field. As an application, we show the existence of ( f, f ′) with ℒ ( f, f ′) ≢ 0.