Associated to each newform $f$ and primitive Dirichlet character $\psi$, there is a unique newform $g:=f\otimes\psi$, the **twist** of $f$ by $\psi$, that satisfies
\[
a_n(g)=\psi(n)a_n(f)
\]
for all integers $n\ge 1$ coprime to $N$ and the conductor of $\psi$. The newforms $f$ and $g$ are then **twist equivalent**. When $g$ is a Galois conjugate of $f$, it is said to be an inner twist.

The newform orbit $[g]$ is a **twist** of the newform orbit $[f]$ by the character orbit $[\psi]$ if some $g\in [g]$ is a twist of $f$ by some $\psi$ in $[\psi]$. This may occur with multiplicity.

Twist equivalence is an equivalence relation. The **twist class** of a newform or newform orbit is its equivalence class under this relation.

In the LMFDB each twist class is identified by the label of its minimal twist.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-02-06 08:03:53

**Referred to by:**

- cmf.minimal_twist
- cmf.twist_minimal
- cmf.twist_multiplicity
- rcs.cande.cmf
- lmfdb/classical_modular_forms/templates/cmf_embedded_newform.html (line 211)
- lmfdb/classical_modular_forms/templates/cmf_newform.html (line 217)
- lmfdb/classical_modular_forms/web_newform.py (line 1011)
- lmfdb/classical_modular_forms/web_newform.py (line 1017)
- lmfdb/classical_modular_forms/web_newform.py (lines 1033-1034)
- lmfdb/classical_modular_forms/web_newform.py (lines 1073-1078)
- lmfdb/classical_modular_forms/web_newform.py (lines 1094-1095)

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

- 2020-02-06 08:03:53 by Andrew Sutherland (Reviewed)
- 2020-01-03 05:50:44 by Andrew Sutherland
- 2020-01-02 21:30:49 by Andrew Sutherland
- 2020-01-02 21:28:01 by Andrew Sutherland

**Differences**(show/hide)