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.
- Review status: reviewed
- Last edited by Andrew Sutherland on 2019-05-10 12:19:16
- lmfdb/classical_modular_forms/main.py (lines 839-844)
- lmfdb/classical_modular_forms/main.py (line 1340)
- lmfdb/classical_modular_forms/main.py (line 1575)
- lmfdb/classical_modular_forms/main.py (line 1583)
- 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