Two (number) fields are arithmetically equivalent if their Dedekind zeta functions are the same. It is known that any two arithmetically equivalent fields are not necessarily isomorphic; Prasad (http://www.math.tifr.res.in/~dprasad/refined-equiv.pdf) gives an example of two non-isomorphic, arithmetically equivalent fields which idele class groups are isomorphic.
Under Langlands Philosophy, the structure of absolute Galois groups of number fields can be understood through $n$-dimensional automorphic representations of adele rings. My question is:
Do there exist two non-isomorphic, arithmetically equivalent number fields which $n$-dimensional automorphic representations of their adele rings are isomorphic? Here, $n>1$ is assumed.