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We present a novel integral representation for a quotient of global automorphic $L$-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. As such, the integral itself constitutes a period, of an arithmetic nature. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over $p$-adic and archimedean fields, and a new Casselman--Shalika type formula.