Let $f$ be a modular form -- more specifically, a normalized new eigenform which is not of CM type.


We say $f$ has **extra twists** if there exists some $\sigma \in \operatorname{Aut}( \mathbf{C})$ such that the Galois conjugate $f^\sigma$ is equal to the twist of $f$ by some non-trivial Dirichlet character, so $a\_n(f)^\sigma = \chi(n) a\_n(f)$ for all $n$ coprime to the level.


Any newform of nontrivial character has at least one extra twist (because the complex conjugate $\bar{f}$ is a twist of $f$.) But browsing some tables suggests that extra twists are pretty unusual for newforms of trivial character.



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> * Do "most" non-CM newforms of trivial character (in some precisely quantifiable sense) have no extra twists?
> * Do "most" newforms of non-trivial character have no extra twists other than the obvious one?
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(This question is prompted by [this earlier question](https://mathoverflow.net/questions/220845/when-is-the-image-of-a-2-dim-l-adic-representation-associated-to-a-modular-form), which is subtle precisely when extra twists exist.)



Do "most" modular forms have no extra twists?
If a newform $f \in \mathcal{S}\_k^{\mathrm{new}}(\Gamma\_0(N),\varepsilon)$ has an inner twist by some $\sigma \in \operatorname{Aut}(\mathbb{C})$, then $f^{\sigma}$ is a newform of the same level as $f$. Moreover, if $\varepsilon$ is trivial, then so is the nebentypus of $f^{\sigma}$ (see (3.8) of [Ribet's paper](https://math.berkeley.edu/~ribet/Articles/annalen_253.pdf)), and so any inner twist must arise from a quadratic Dirichlet character.


Ribet's paper actually shows (see (3.9)) that if $N$ is squarefree, then there are no nontrivial inner twists. More precisely, he shows that if $N$ is squarefree and $\chi$ is a quadratic Dirichlet character, then the twist $f \otimes \chi$ cannot have level $N$ and trivial nebentypus.


This is not hard to see by a local argument: if $N$ is squarefree, then for any $p \mid N$, the local component $\pi\_p$ of the associated automorphic representation must be isomorphic to $\mathrm{St}$, the Steinberg representation of conductor $p$ associated to the trivial character. Any nontrivial twist of $f$ leaving the nebentypus unchanged must necessarily be quadratic, but any twist of $\mathrm{St}$ by a ramified quadratic character has conductor at least $p^2$, not $p$ (and in fact exactly $p^2$ if $p$ is odd). Similarly, at any unramified place, the twist of an unramified principal series representation by a ramified quadratic character has conductor at least $p^2$. So the level of $f \otimes \chi$ is strictly greater than $N$.


But if $N$ is not squarefree, then this is no longer the case! Indeed, in a [recent paper](http://arxiv.org/pdf/1502.06885.pdf) (Theorem 6.4), I proved the following result (well, technically I proved it for Maaß cusp forms, but it generalises in an obvious way to holomorphic cusp forms).



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> Fix a nonsquarefree odd integer $N$, and let $N' > 1$ be squarefree and such that $N'^2 \mid N$. Let $\mathcal{S}\_k^{\mathrm{new}}(\Gamma\_0(N))\_{\mathrm{nonmon}(\varepsilon\_{\mathrm{quad}(N')})}$ denote the vector space spanned by newforms $f$ of weight $k$, level $N$, and trivial nebentypus such that $f$ does not have CM by the quadratic Dirichlet character $\varepsilon\_{\mathrm{quad}(N')}$ modulo $N'$, but that the twist $f \otimes \varepsilon\_{\mathrm{quad}(N')}$ is also a newform of level $N$ and trivial nebentypus. Then
>  \[\frac{\dim \mathcal{S}\_{k}^{\mathrm{new}} (\Gamma\_0(N))\_{\mathrm{nonmon} (\varepsilon \_{\mathrm{quad}(N')} )} }{\dim \mathcal{S}\_k^{\mathrm{new}}(\Gamma\_0(N))} \sim \prod\_{\substack{p \mid N' \\ p^2 \parallel N}} \left(1 - \frac{p}{p^2 - p - 1}\right)\]
>  as $k$ tends to infinity over the even integers.
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> Furthermore, this still holds if we replace $\mathcal{S}\_k^{\mathrm{new}}(\Gamma\_0(N))\_{\mathrm{nonmon}(\varepsilon\_{\mathrm{quad}(N')})}$ with
>  \[\bigcap\_{\substack{N^\* \mid N' \\ N^\* > 1}} \mathcal{S}\_k^{\mathrm{new}}(\Gamma\_0(N))\_{\mathrm{nonmon}(\varepsilon \_{\mathrm{quad}(N^\*)})}.\]
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A similar result holds even when $N$ is squarefree if $\varepsilon$ is nontrivial; see Proposition 6.5 of my paper.


That inner twists occur in abundance when $N$ is not squarefree was observed as far back as 1977 by Ribet; at the bottom of page 48 of [this paper](https://dx.doi.org/10.1007/BFb0063943), Ribet writes 



> 
> It would be of interest to give an *a priori* construction of forms
>  with extra twists. If the level $N$ is divisible by a high power of a
>  prime, these forms seem to be more the rule than the exception.
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