Let $\mbox{Ex}(N) = \{ e_{0} =1, e_{1}, e_{2}, \cdots \}$ be a group of exact divisors of $N$ under the opreration $e * e’ = ee’ / \gcd(e,e’)^2$.
By a sign pattern $\varepsilon$ for $N$ is meant a group homomorphism
$$\varepsilon : \mbox{Ex}(N) \to \{\pm 1\}, \qquad \varepsilon(1) =1.$$
For a sign pattern $\varepsilon$ of $N$, let $M_{k}^{\varepsilon}(N)$ (resp. $S_{k}^{\varepsilon}(N)$) be a subset of $M_{k}(N)$ (resp. $S_{k}(N)$) consisitng of all forms $f$ such that the Atkin-Lehner eigenvalue of $f$ is $\varepsilon(p)$ for every $p |N$, that is
\begin{eqnarray*}
M_{k}^{\varepsilon}(N) &:=& \{ f \in M_{k}(N) : f \mid_{k} W_{p} = \varepsilon (p) f \quad \mbox{for every prime}~ p|N\}, \cr
S_{k}^{\varepsilon}(N) &:=& \{ f \in S_{k}(N) : f \mid_{k} W_{p} = \varepsilon (p) f \quad \mbox{for every prime}~ p|N\}.
\end{eqnarray*}
Similarly we can define $M_{k}^{!,\varepsilon}(N)$ by the subspace of the weakly holomorphic modular form space $M_{k}^{!}(N)$ with a sign pattern $\varepsilon$.
Let $m_{N,k}^{\varepsilon} := \max\{ \mbox{ord}_{\infty} f : f \in M_{N,k}^{\varepsilon}\}$. Then the form $f$ in $M_{N,k}^{\varepsilon}$ such that $\mbox{ord}_{\infty} f = m_{N,k}^{\varepsilon}$ uniquely exists.
We denote such $f$ by $\Delta_{N,k}^{\varepsilon}$, that is $\Delta_{N,k}^{\varepsilon}$ is the unique form in $M_{N,k}^{\varepsilon}$ with maximal order of vanishing at infinity.
We exhibit the recipe for construct $\Delta_{N,k}^{\varepsilon}$ and its $q$-expansion : Download the data ;
The data files are sorted by space level and each level has a separate folder. In each level’s folder, there is a separate text file for each possible sign pattern at that level and the weight required for basis construction. These text files provide information on how to find the delta and its q-expansion.
The file names follow the convention D{level}{sign pattern}{weight}.
For instance, the file named “D102pmm2” in the N=102 folder deals with finding $\Delta_{102,2}^{(+,-,-)}$ for level $102$, sign pattern $(+,-,-)$, and weight $2$.
Last modified : 2023. 3. 3.