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Does this mean it's dimension is infinite?

Another question:

I am trying to find a basis for M2!, denoting the space of modular forms of weight -infinity < k < 2 but non-holomorphic with a simple pole at infinity of order 1.

But from the valence formula, the $k/12$ is negative and all other terms will be negative, except a positive contribution of 1 from the pole at infinity, so does this imply that the dimension of M_k for all $k$ between $-\infty <0 < 2$ is zero except for $k=-12$.

Thanks.

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    You may want to take a peek at http://math.mit.edu/~brubaker/Math784/spacesofmodularforms.pdf – 2017-01-04
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    The dimension of the space of holomorphic modular forms (ie, holomorphic on the upper half plane AND at all cusps), as a complex vector space, is 1, given by the constants. – 2017-01-04

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