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It is well known that the algebra of modular forms of integral weight, trivial multiplier and $\mathrm{SL}(2,\mathbb{Z})$ is generated by $E_4$ and $E_6,$ and is isomorphic to the following weighted polynomial ring $$ \mathbb{C}[E_4,E_6] $$ with $\mathrm{wt}(E_4)=4$ and $\mathrm{wt}(E_6)=6.$ I would like to know other algebras of modular forms, ie., ones of other groups, other multipliers and other weight types (half-integral weight, third-integral weight and so on).

Besides, I would like to konw Hauptmoduln of various groups (the Hauptmodul of $\mathrm{SL}(2,\mathbb{Z})$ is Klein's j-invarant).

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    You can't add modular forms of different weight, so $\mathbb{C}[E_4,E_6]$ is the ring generated by the modular forms, but doesn't contain only them. In [Diamond & Shurman](http://148.206.53.84/tesiuami/S_pdfs/109105%20A%20first%20course%20in%20modular%20forms%2008%20zal.pdf) (first example p.86) $A_0(\Gamma_0(n))$ the field of modular functions (weight $0$) for $\Gamma_0(n)$ is studied in details, $A_0(\Gamma_0(n))$ being a finite extension of $A_0(\Gamma_0(1))$ – 2017-01-27
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    What I wrote was not correct. It is true that $E_4+E_6$ is not a modular form, for example. I would like to know the structure of graded ring $\oplus_{n\in \mathbb{Z}}\mathcal{M}_{\alpha n}(\Gamma,v^n)$ with $\Gamma\subset\mathrm{SL}(2,\mathbb{Z})$ a subgroup, $\alpha\in\mathbb{R}_{>0}$ and $v$ a multiplier of weight $\alpha$. Diamond & Shurman is one of what I would like to know. Thank you for informing me. – 2017-01-28

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