Oops, I see there was a typo. I meant Greek!

@Pietro: I'm just quoting the source I referenced. Don't blame the messenger!

I looked up Shimura's book. It was a sixth book that someone had checked out. It only covers the cuspidal case.

Greg, some people have communicated with Gauthier, but I certainly agree that this mathoverflow post is not generating anything productive anymore.

The basic reason is that the cocycle you mentioned, integrating from $i$ to $\gamma\cdot i$, vanishes on $S$, which means it can't be nonzero for a cuspidal cycle, which vanishes on the other generator $T$.

Shimura has written several books. There were at least five in our library and a cursory glance did not reveal anything to do with cocycles or cohomology. I agree that taking any $z$ in the upper half plane and integrating $z$ to $\gamma z$ will give a cocycle for any modular form, but you can't get enough of them this way. To get most of them, you have to start with a point on the boundary, such as $0$, but this construction only seems to be well-defined for cuspidal forms.

@Kevin: which book of Shimura are you referring to?

I know that for cuspidal forms $f(z)$ you integrate $f(z)z^s$ from $0$ to $\gamma 0$ to get a cocycle. I guess you're saying I should do a similar thing, but not use a boundary point.

Yeah, maybe changing the font is the simplest answer.