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 $\psi$. If the newform $g$ 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.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by Andrew Sutherland on 2020-01-03 05:50:44

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

- 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

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