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We consider specific linear combinations of two loop modular graph functionson the toroidal worldsheet with $2s$ links for $s=2, 3$ and $4$. In each case,it satisfies an eigenvalue equation with source terms involving $E_{2s}$ and$E_s^2$ only. On removing certain combinations of $E_{2s}$ and $E_s^2$ from it,we express the resulting expression as an absolutely convergent Poincareseries. This is used to calculate the power behaved terms in the asymptoticexpansion of the zero mode of the Fourier expansion of these graphs in a simplemanner.

Zero mode of the Fourier series of some modular graphs from Poincare series