The secret inner meaning of modular forms

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SeqFans,

John Baez posted the following to NG sci.math.research:
<<
Betreff: The secret inner meaning of modular forms
Datum:   Wed, 7 Jan 2004 03:07:04 +0000 (UTC)
Von:     baez at math.removethis.ucr.andthis.edu (s.p.research moderator)
Firma:   UCR
Foren:   sci.math.research

In number theory people like to count things, and this gives
sequences of numbers a_n where n = 0,1,2,....  Then, they like
to package these sequences as generating functions

f(q) = sum a_n q^n  

And then, they get excited when these generating functions are
"modular forms", or more generally, when they transform in some 
nice simple way under

z -> 1/z

where

q = exp(2 pi i z)

My question is: what does it really MEAN about a sequence of
numbers a_n when the generating function f is a modular form, 
or transforms simply under z -> 1/z?  

The basic textbooks I've been reading tend to focus on the cool 
things you can do when f is a modular form - not on how you can 
stare at a sequence a_n and tell when f is a modular form.  

So: what's the secret inner meaning of modularity, THOUGHT OF AS
A PROPERTY OF A SEQUENCE a_n?  

There should be some simple conceptual answer, no?
>>

Any comments? And please indicate if it's ok to forward answers to
the (moderated) NG sci.math.research. I would then remove the e-mail
address. Or better: Post to the NG yourself.

Hugo Pfoertner