A newform $f$ admits a **self-twist** by a primitive
Dirichlet character $\chi$ if the equality
\[
a_p(f) = \chi(p)a_p(f)
\]
holds for all but finitely many primes $p$.

For non-trivial $\chi$ this can hold only when $\chi$ has order $2$ and $a_p=0$ for all primes $p$ not dividing the level of $f$ for which $\chi(p)=-1$. The character $\chi$ is then the Kronecker character of a quadratic field $K$ and may be identified by the discriminant $D$ of $K$.

If $D$ is negative, the modular form $f$ is said to have complex multiplication (CM) by $K$, and if $D$ is positive, $f$ is said to have real multiplication (RM) by $K$. The latter can occur only when $f$ is a modular form of weight $1$ whose projective image is dihedral.

It is possible for a modular form to have multiple non-trivial self twists; this occurs precisely when $f$ is a modular form of weight one whose projective image is isomorphic to $D_2:=C_2\times C_2$; in this case $f$ admits three non-trivial self twists, two of which are CM and one of which is RM.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2019-05-10 12:19:16

**Referred to by:**

- cmf.inner_twist
- cmf.inner_twist_count
- cmf.inner_twist_group
- cmf.nontrivial_twist
- cmf.rm_form
- cmf.self_twist_field
- rcs.cande.cmf
- rcs.rigor.cmf
- lmfdb/classical_modular_forms/main.py (lines 815-820)
- lmfdb/classical_modular_forms/main.py (line 1304)
- lmfdb/classical_modular_forms/main.py (line 1538)
- lmfdb/classical_modular_forms/main.py (line 1546)

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

- 2019-05-10 12:19:16 by Andrew Sutherland (Reviewed)
- 2019-03-21 13:58:45 by John Cremona (Reviewed)
- 2018-12-05 21:21:12 by Andrew Sutherland

**Differences**(show/hide)