In this paper we will consider Fourier-Jacobi expansions of nearly holomorphic modular forms on , whose coefficients are nearly holomorphic theta functions, and will construct a certain Galois action on them.
Atsuo YAMAUCHI. "A certain Galois action on nearly holomorphic modular forms with respect to a unitary group in three variables." J. Math. Soc. Japan 55 (4) 995 - 1031, October, 2003. https://doi.org/10.2969/jmsj/1191418761