---
date: 2023-03-04 22:37
aliases: ["quasimodular", quasimodular forms]
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Last modified: `=this.file.mday`

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [modular form](Unsorted/modular%20form.md)

---

# quasimodular

![](attachments/2023-03-04modf.png)

![](attachments/2023-03-04wme.png)

![](attachments/2023-03-04asdqwe.png)

![](attachments/2023-03-04triv.png)

![](attachments/2023-03-04eisen.png)

Dimensions of spaces of modular forms:

$\operatorname{dim} \mathcal{M}_k\left(\mathrm{SL}_2(\mathbb{Z})\right)=\left\lfloor\frac{k}{12}\right\rfloor+ \begin{cases}1 & \text { for } k \equiv 0,4,6,8,10 \quad(\bmod 12), \text { and } \\ 0 & \text { for } k \equiv 2 \quad(\bmod 12)\end{cases}$

Ramanujan's form: $\Delta(z)=\frac{E_4(z)^3-E_6(z)^2}{1728}=q \prod_{n=1}^{\infty}\left(1-q^n\right)^{24}$, a weakly modular form for $\SL_2(\ZZ)$. Used to cancel poles at cusps.

Graded ring of weakly modular forms:
$\mathcal{M}^{!}\left(\mathrm{SL}_2(\mathbb{Z})\right)=\mathbb{C}\left[E_4, E_6, \Delta^{-1}\right]$


## Quasimodularity

![](attachments/2023-03-04example.png)

![](attachments/2023-03-05differentations.png)