The article generalizes an observation of Zagier and Gangl to show that the image of the spectral Eisenstein series on a general congruence subgroup of \(\text {SL}_2(\mathbb {Z})\) , under the Eichler–Shimura isomorphism, is defined over a cyclotomic number field. We use the same technique to generalize an invariant attached to imaginary quadratic fields in connection with the polylogarithm conjecture on the special values of L-functions. Our treatment also provides an elementary derivation of the Fourier expansion of the Maass Eisenstein series on congruence subgroups presented as a power series.
