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We define canonical real analytic versions of modular forms of integralweight for the full modular group, generalising real analytic Eisensteinseries. They are harmonic Maass waveforms with poles at the cusp, whose Fouriercoefficients involve periods and quasi-periods of cusp forms, which areconjecturally transcendental. In particular, we settle the question of findingexplicit `weak harmonic lifts' for every eigenform of integral weight $k$ andlevel one. We show that mock modular forms of integral weight arealgebro-geometric and have Fourier coefficients proportional to $n^{1-k}(a'_n +\rho a_n)$ for $n\neq 0$, where $\rho$ is the normalised permanent of theperiod matrix of the corresponding motive, and $a_n, a'_n$ are the Fouriercoefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform,respectively. More generally, this framework provides a conceptual explanationfor the algebraicity of the coefficients of mock modular forms in the CM case.

A class of non-holomorphic modular forms III: real analytic cusp forms for $\mathrm{SL}_2(\mathbb{Z})$