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@paulgarrett: Thanks for your response! Is there a way to begin with a finite codim ideal $I$ in $\mathcal{Z}$ and construct an Eisenstein series annihilated by $I$? I'm mostly interested in whether or not a characterization of the annihilators of automorphic forms exists that distinguishes them in the finite codimensional ideals of $\mathcal{Z}$, and (better yet!) characterizations that allow for distinguishing annihilators of automorphic forms of a fixed growth rate.
