# Modular Forms, Dimension, Valence Formula

1. Apr 17, 2017

### binbagsss

1. The problem statement, all variables and given/known data

What is the dimension of $M_{24}$?

2. Relevant equations

attached
3. The attempt at a solution

I am confused what the (mod 12) is referring to- is it referring to the $[k/12]$ where the square brackets denote an equivalent class and the $k \equiv 2$ / $k \notequiv 2$ or just the $[k/12]$?

I am confused because I thought $k \equiv 2$ (mod 12) only when $k=24$, so for the dimension $M_2$ we would need to look at the top definition, however clearly the bottom has been used, which makes me think that the '(mod $12$)' is only referring to the square brackets?

In which case for $M_{24}$ I need to look at the top line and conclude $dim M_{24}=3$, however if (mod 12) is referring to both then I need to look at the bottom line and conclude $dim M_{24}=2$, however in this case it makes no sense how we have got $dim M_2=0$

Thanks .

2. Apr 17, 2017

### Staff: Mentor

$24 \equiv 0 \operatorname{mod}12$
$2 \equiv 2 \operatorname{mod}12$

3. Apr 20, 2017

### binbagsss

So
$4 \equiv 4 \operatorname{mod}12$

So from the definition above $dim M_{4} =[k/12]=[4/12]$;

how is $[4/12]$ 1? Isn't this zero too? what do the square brackets denote.

E.g $14\equiv 2$ (mod 12) so am I using the original $k$ : $[14/12]$ or $[2/12]$?

4. Apr 20, 2017

### binbagsss

Oh it doesn't matter, [ ] denote equivalent classes, so it's 'the remainder of the division' which is $2$ in both of these cases?

Can I just test my understanding here- is $dim M_{28}=[k/12]+1=5$?

Last edited: Apr 20, 2017
5. Apr 20, 2017

### binbagsss

No I'm lost $dim M_{12}=2$ but I am getting:

$12 \equiv 0$ mod 12, so I'm looking at $[k/12]+1$, if these [ ] denote equivalent classes a number divisible by 12 is represented by the element $0$ so I get $0+1=1$...

Unless these [ ] square brackets denote taking the integer or something? what do these square brackets mean? thanks.

6. Apr 20, 2017

### Dick

The brackets are the floor function. The greatest integer less than or equal to the quotient.