Is there any construction of Abelian variety associated to Hilbert modular forms with arbitrary nebentypus? I am aware of the Sho-Wu Zhang's 2001 Annals paper for $\Gamma_0(N)$ under Jacquet-Langlands condition. Is there any generalization of the construction for arbitrary nebentypus under certain Jacquet Langlands condition and restriction on the degree of the field in question.
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