# Modular form on $\Gamma_0(N)$

I recently asked this question on Math.StackExchange with no answer so far. So I thought maybe I can find an answer here.

Let $M(k,\Gamma_0(N))$ be a space of modular forms of weight $k$ on $\Gamma_0(N)$. Each $f \in M(k,\Gamma_0(N))$ has a Fourier expansion of the form $$f(\tau)=\sum_{n\in \mathbb{N}}a(n)\,q^n \quad \text{where}\quad q=\mathrm{e}^{2\pi \mathrm{i}\tau} .$$

Now, let $g(\tau)$ be a function obtained from $f(\tau)$ by omitting all $a(n)$ such that $\gcd(n,N)\neq 1$, i.e. $$g(\tau)=\sum_{\substack{n\in \mathbb{N}\\(n,N)=1}}a(n)\,q^n .$$

Question: Is $g(\tau)$ a modular form? What is its level?

Many thanks.

• Dear @user31009, perhaps you should wait at least a few days (instead of 20 hours) before reposting your Math.StackExchange question on MathOverflow. Quick reposting leads to duplication of effort, and is frowned upon by both communities. Feb 21, 2014 at 19:44
• Not to mention that questions can be migrated. You can flag a question for migration by a moderator, in which case people can follow the link to the other site, and there won't be duplication. Feb 21, 2014 at 20:42
• I think this is a perfectly good question and should not be closed. Feb 21, 2014 at 23:54

$$g(\tau)=f(\tau)\otimes \left(\tfrac{N^2}{\cdot}\right)=\sum_{n\in \mathbb{N}}\left(\tfrac{N^2}{n}\right)a(n)\,q^n \quad \text{where}\left(\tfrac{N^2}{\cdot}\right) \text{ is the Kronecker symbol}.$$ This means that $g(\tau)$ is just the twist of $f(\tau)$ by a principle character. Indeed we have $$g(\tau)\in M(k,\Gamma_0(N^3)).$$
We can express the form $g(\tau)$ as $$g(\tau)=\sum_{d\mid N}\mu(d)\sum_{\substack{n\in\mathbb{N}\\d\mid n}} a(n)\,q^n.$$ In the notation of Atkin-Lehner (Hecke operators on $\Gamma_0(m)$, Math. Ann. 185 (1970), 134-160), the inner sum is $(f\mid U_d)\mid B_d$, which lies in $M(k,\Gamma_0(dN))$ by Lemmata 2, 6, 14 in the paper. This implies that $$g\in M(k,\Gamma_0(N^2)).$$
• This seems like a bit more generality than necessary - doesn't the modularity follow straightforwardly from properties of $U_p$ and $V_p$? Feb 21, 2014 at 21:15