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A newform $f=\sum a_n q^n$ admits a self twist by a nontrivial primitive Dirichlet character $\chi$ if we have $a_p=\chi(p)a_p$ for all but finitely many primes $p$.

This can occur 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 more than one self twist; 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 self twists, two of which are CM and one of which is RM.

Knowl status:
  • Review status: beta
  • Last edited by Eran Assaf on 2023-06-30 19:47:50
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