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We begin a study of $m$th Chern classes and $m$th characteristic symbols for Iwasawa modules which are supported in codimension at least $m$. This extends the classical theory of characteristic ideals and their generators for Iwasawa modules which are torsion, i.e., supported in codimension at least $1$. We apply this to an Iwasawa module constructed from an inverse limit of $p$-parts of ideal class groups of abelian extensions of an imaginary quadratic field. When this module is pseudo-null, which is conjecturally always the case, we determine its second Chern class and show that it has a characteristic symbol given by the Steinberg symbol of two Katz $p$-adic $L$-functions.