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2021 | OriginalPaper | Chapter

# On Rational Bianchi Newforms and Abelian Surfaces with Quaternionic Multiplication

Authors : J. E. Cremona, Lassina Dembélé, Ariel Pacetti, Ciaran Schembri, John Voight

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## Abstract

We study the rational Bianchi newforms (weight 2, trivial character, with rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic curves, but instead to abelian surfaces with quaternionic multiplication. Two of these examples exhibit a rather special kind of behaviour: we show they arise from twisted base change of a classical newform with nebentypus character of order 4 and eight inner twists.
Footnotes
1
Here, 2.​0.​7.​1 is the LMFDB label for the base field $$K=\mathbb {Q}(\sqrt {-7})$$ and 30625.1 the label for the level ideal (175), which has norm 30625. The final c is the alphabetic label for this specific newform at that level. We use either full labels such as 2.​0.​7.​1-30625.​1-c for Bianchi newforms, or the shorter version 30625.​1-c which omits the field when that is clear from the context.

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Title
On Rational Bianchi Newforms and Abelian Surfaces with Quaternionic Multiplication
Authors
J. E. Cremona
Lassina Dembélé
Ariel Pacetti
Ciaran Schembri
John Voight