Galois conjugate newforms $f$ and $g$ are inner twists if there is a Dirichlet character $\chi$ such that \[ a_p(g) = \chi(p)a_p(f) \] for all but finitely many primes $p$. Without loss of generality, we may assume that $\chi$ is a primitive Dirichlet character, and by a theorem of Ribet [MR:0594532, 10.1007/BF01457819], the newform $g$ is conjugate to $f$ via a $\Q$-automorphism $\sigma$ of the coefficient field of $f$. The set of pairs $(\chi,\sigma)$ form the group of inner twists of $f$.
Each pair $(\chi,\sigma)$ corresponding to an inner twist of $f$ is uniquely determined by the the primitive character $\chi$, and we say that $f$ admits an inner twist by $\chi$. When $\sigma=1$ is is the trivial automorphism, we have $g=f$ and say that $f$ admits a self twist by $\chi$; in this case $\chi$ is either the trivial character or the Kronecker character of a quadratic field.
The home page of each newform in the LMFDB includes a list of inner twists, in which non-trivial self twists are distinguished by listing the associated quadratic field (the CM or RM field), while inner twists that are not self twists are simply marked as "inner".
- Review status: reviewed
- Last edited by Andrew Sutherland on 2021-05-07 16:55:02
- lmfdb/classical_modular_forms/main.py (line 1342)
- lmfdb/classical_modular_forms/templates/cmf_newform.html (lines 209-211)
- 2021-05-07 16:55:02 by Andrew Sutherland (Reviewed)
- 2020-01-02 08:03:52 by Andrew Sutherland (Reviewed)
- 2019-03-08 15:43:29 by Andrew Sutherland (Reviewed)