Twisting modular forms by Dirichlet characters

Question

Let $\chi,\chi_1$ be Dirichlet characters modulo $M$ and $N$.

In Koblitz's book "Introduction to Elliptic Curves and Modular Forms", Proposition III.3.17, it is proved that if $$f(q)=\sum_{n=0}^\infty a_n q^n$$ is the Fourier expansion of a modular form in $M_k(\Gamma_0(M),\chi)$, then $$f_1(q)=\sum_{n=0}^\infty \chi_1(n) a_n q^n$$ is a modular form in $M_k(\Gamma_0(MN^2),\chi\chi_1^2)$.

But it seemed that one can use Hecke operators to find a stronger result (See the answers of this question https://mathoverflow.net/questions/158278/modular-form-on-gamma-0n).

Is there any better known results about twisting with characters in general or in some special cases?

Bests.

Answer 1

Apparently this question is still "active". As David Loeffler comments, Atkin and Li addressed this in their "Twists of newforms" paper (Prop 3.1), which says the following.

Let $\epsilon$ be a character of conductor $N'$ and $\chi$ a character mod $M$. Then for $f \in M_k(N, \epsilon)$, the twist of $f$ by $\chi$ lives in $M_k(N'', \epsilon \chi^2)$, where $N''$ is the lcm of $N, N'M$ and $M^2$.

This is as good as you can get in general, though for some forms, the "exact level" will be lower than this. E.g., say $M$ and $N$ are coprime, $g \in M_k(N)$ and $\chi$ is a character of conductor $M$. If $f$ is the twist of $g$ by $\chi$, it will have level $M^2N$, and if you twist $f$ by $\chi^{-1}$ you get back $g$, i.e., you lower the level from $M^2 N$ to $N$ by twisting. Intermediate situations can also happen.

So you can't say the exact level of a twist just given the level and conductors of characters in general, but under some additional conditions you can--see Thm 3.1 of Atkin and Li.

Note Hijikata, Pizer and Shemanske also have a paper on "Twists of newforms", where they get similar results by different methods, and moreover show that one can decompose certain new spaces into direct sums of twists of other new spaces. (In particular this tells you the exact level of various twists, though perhaps not under any more general conditions than what Atkin and Li do.)