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How can I prove that:

  • For a given natural number $k$ the dimension space of modular forms of weight $2k$ is $\lfloor{\frac{k}{6}\rfloor}+1$ if $k \not\equiv 1 \: (\text{mod}\ 6)$ and $\lfloor{\frac{k}{6}\rfloor}$ if $k \equiv 1 \: (\text{mod}\ 6)$
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This presumably means of level 1. Where are you learning about modular forms? Most introductory books that have this statement also prove it... – Matt Apr 28 '12 at 19:59

This is a very standard argument, see e.g. Corollary 2.16 of William Stein's book "Modular Forms: A Computational Approach". This is available as a free e-book; the relevant section is here:

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