Suppose I have a holomorphic modular form $f \in M_k(\Gamma_0(N), \chi)$ with $k \in \mathbb Z^+$ and $f = \sum_{n=1}^\infty a(n)q^n$. Considering the $q$-expansion, one may suspicious that $f$ is actually a cusp form. Then, it remains to check that $f$ vanishes at the cusps of $\Gamma_0(N)$ (say $\Gamma_0(p)$, so we only have the two cusps).

It is clear from the $q$-expansion of $f$ that it vanishes at the cusp at $\infty$, but how exactly do you show that it vanishes at the cusp at $0$?

I have heard that it has something to do with finding $\gamma = \left( \begin{array}{c c} a & b \\ c & d \end{array} \right) \in GL_2(\mathbb Z)^+$ that takes $\infty$ to $0$ under the transformation $\gamma z = \frac{az+b}{cz+d}$. Then, somehow wrangle $f |_k \gamma = (cz+d)^{-k} f(\gamma z)$ back into a $q$-expansion at $\infty$. If this has no constant term, then $f$ vanishes at the cusp at $0$.

If this is indeed the procedure for verifying vanishing at the cusps, then it is unclear to me what sort of "wrangling" needs to occur. I think the Fricke involution $W_N = \left( \begin{array}{c c} 0 & -1 \\ N & 0 \end{array} \right)$ would work for $\gamma$, but the $q$-expansion of $f$ is an unintelligible mess after hitting it with the $|_k$ operator.


As far as I can tell, the method listed in the question has little practical value. Mangling the $q$-expansion with the $|_k$ operator makes it very difficult to get back to a $q_N$ expansion.

A more hopeful approach would be to attempt to create a basis-like set of modular forms for $S_k(\Gamma_0(N), \chi)$ and then to write $f$ as a linear combination of these forms.

For example, in Arithmetic of the 13-regular partition function modulo 3, Webb uses Dedekind's eta function $$ \eta(z) := q^{1/24} \prod_{n=1}^\infty (1-q^n) $$ to create a pseudo-basis for $M_{12}(\Gamma_0(312), \chi_{13})$ modulo 3. Many of the forms in his "basis" are actually cusp forms, and could potentially be used create the desired cusp form (modulo 3) in $M_{12}(\Gamma_0(312), \chi_{13})$.


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