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Suppose one was given a sequence and $a_{0}, a_{1}, a_{2}, \ldots$. Is there a searchable database somewhere to see if $a_{0} + a_{1}q + a_{2}q^{2} + \cdots$ is expressible as modular form (or some product of modular forms, etc.)?

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Have you looked at ? I'm not really familiar with this reference, but it looks relevant. – David Speyer Jun 11 '11 at 21:47
I asked this once. I don't remember exactly how I phrased it, so this answer might not be true. I was told that unless I really had infinitely many coefficients I couldn't tell. If you can pin down a weight and level (maybe only one of those is needed), this is what allows you to only have to check a finite number of coefficients. – Matt Jun 17 '12 at 15:53
up vote 4 down vote accepted

There are two databases that contain some known modular forms. One is Stein's and this is the other.

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