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 number of inner twists of $f$ is an invariant of its Galois orbit, as is the number of inner twists by characters in any particular Galois orbit of Dirichlet characters.

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".

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2021-05-07 16:55:02

**Referred to by:**

- cmf.inner_twist_count
- cmf.inner_twist_group
- cmf.inner_twist_multiplicity
- cmf.inner_twist_proved
- cmf.nontrivial_twist
- cmf.self_twist_field
- cmf.twist
- cmf.twist_multiplicity
- rcs.cande.cmf
- rcs.rigor.cmf
- lmfdb/classical_modular_forms/main.py (line 1342)
- lmfdb/classical_modular_forms/templates/cmf_newform.html (lines 209-211)

**History:**(expand/hide all)

- 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)

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