Twisting modular forms by Dirichlet characters

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.

• Your question is rather vague. What exactly is it that you want to ask? Are you asking whether the $MN^2$ is optimal? (It always is when $M$ and $N$ are coprime). Commented Aug 13, 2014 at 0:49
• @DavidLoeffler thanks for your comment. For $M$ and $N$ not coprime, is the level of the twisted version can be smaller than $MN^2$? if yes, can you please give me more details? Bests.
– user80225
Commented Aug 13, 2014 at 1:15
• For a definitive statement of what one can prove by purely elementary methods, see the article "Twists of newforms and pseudo-eigenvalues of W" by Atkin and Li, 1973 or thereabouts. Commented Aug 13, 2014 at 14:49

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.