For compact \(J\subset{\mathrm{GL}}_n(F)\), the homomorphism \(v_F\circ\det\) has finite image in \(\mathbb Z\), hence image zero. An unramified character
is therefore trivial on \(J\). The compact packet restrictions of \(\pi\) and \(\pi\otimes\nu_a\) agree, while the central character changes by \(a^n\) on a scalar uniformizer. Choosing \(a^n\ne1\) gives a different atom.
Block 167 also records a rank-eight skipped-degree \(8\to2\) host with the same torsion, Swan, and self-pair conductor as the full tower, but quotient dimension 4 rather than 6.
Reading the diagram
The finite packet, numerical profiles, and noncompact normalization contain different information. No one of these alone is a substitute for an arithmetic witness preserving the full declared datum.