On coefficients of Poincaré series and single-valued periods of modular forms

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level $$ and integral weight $$ coincides with the field generated by the single-valued periods of a certain motive attached to $$ . This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres-Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann-Ono's construction of harmonic lifts of Poincaré series.