Modular form on $\Gamma_0(N)$

Question

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.

Answer 1

$$ g(\tau)=f(\tau)\otimes \left(\tfrac{N^2}{\cdot}\right)=\sum_{n\in \mathbb{N}}\left(\tfrac{N^2}{n}\right)a(n)\,q^n \quad \text{where}\left(\tfrac{N^2}{\cdot}\right) \text{ is the Kronecker symbol}.$$ This means that $g(\tau)$ is just the twist of $f(\tau)$ by a principle character. Indeed we have $$g(\tau)\in M(k,\Gamma_0(N^3)).$$

Answer 2

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 $$ g\in M(k,\Gamma_0(N^2)). $$

Answer 3

The answer is yes it is a modular form. For this (and the level) see Section 3.8 of Soma Purkait's thesis.