# On rational Bianchi newforms and abelian surfaces with quaternionic multiplication

## TL;DR

This paper investigates rational Bianchi newforms linked to abelian surfaces with quaternionic multiplication, revealing special cases arising from twisted base change of classical newforms with specific characters.

## Contribution

It identifies and analyzes rational Bianchi newforms associated with abelian surfaces with quaternionic multiplication, highlighting their origin from twisted base change of classical newforms.

## Key findings

- Two examples exhibit unique behavior from twisted base change.
- Rational Bianchi newforms can be connected to classical newforms with specific characters.
- The study expands understanding of the relationship between Bianchi newforms and abelian surfaces.

## 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.