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Excuse me this is probably a really stupid question but I ask because I thought that the definition of the dimension of a space is the number of elements in the basis.

Now I have a theorem that tells me that

## dim M_{k} = [k/12] + 1 if k\neq 2 (mod 12)

=[k/12] if k=2 (mod 12) ##

for ## k >0 even ##

So e.g ## dim (M_4) = dim (M_10) = etc =dim(14) ## and ## dim(M_{12})=2 ##

Where ## M_{k} ## denotes a modular form of weight k

I also have proposition ( ring of mod forms) that: ## E_4 ## & ## E_6 ## the Eisenstein series form a basis for ## M_{k} ##, that is every modular form can be written uniquely as a polynomial in ## E_4 ## & ## E_6 ##

My question

If ## E_4 ## & ## E_6 ## are a basis for the modular form then isn't this saying that the dimension of any modular form is 2?

So ## dim (M_k) ## is the number of unique matrices with given weight ## k ## in ## SL_2(z) ## right? So Obviously from the above this isn't correct, so I'm unsure on what the definition of 'basis' is here or...?

Many thanks in advance

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# A Modular forms, dimension and basis confusion, weight mod 1

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