Implemented an Explicit Dimension Formula for the Newspace with character

[eross156]

This PR implements a recently computed explicit dimension formula for newspaces of modular forms with character.
This completes Issue #38384
The tests verify this formula against the current dimension computation in Sage.

I added some additional documentation underneath dimension_new_cusp_forms() for this change. But I can't view the documentation preview, because for some reason, I can't get the docs to build on my machine.

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

⌛ Dependencies

[tscrim]

Thank you for your contribution. @JohnCremona, perhaps you might have some thoughts on this PR?

For me, I would prefer not to have all of those internal functions as they make the code harder to follow (i..e, take a more iterative style approach instead of functional) as most of those are called in exactly one place. You are also going to get a lot of Python ints being mixed in instead of Sage Integers too.

[JohnCremona]

I will take a look -- saw the preprint but have not yet read it as I'm at ANTS this week and then on holdiday for a few days.

[eross156]

@tscrim Thank you for all your feedback and updates. Especially thank you for all the style updates to match this repo; I'm still trying to figure out all the code standards here.

[eross156]

For me, I would prefer not to have all of those internal functions.

We could potentially combine all the func's with the func_local's. (The original thought was to use func_local's, so that I could use early return and cut out some if / else blocks. But this turned out to not be a big deal either way.) Additionally c3, c4, c0 could be converted to just inline expressions. This would cut out about about half of the internal functions. But if you mean cutting out all of the internal functions, I guess I respectfully disagree. I don't know how I would be able to keep everything straight otherwise, comparing with the published formula. But I'm happy to conform to whatever the standards are here. Compare this with the current dimension formula for trivial character in Sage (which the character formula is a generalization of).

You are also going to get a lot of Python ints being mixed in instead of Sage Integers too.

I agree this is weird. Is it considered better practice to change every int literal in the code to e.g. ZZ(999)? Also compare this with the current dimension formula for trivial character in Sage

[tscrim]

Sorry for the delay in responding (traveling).

Yes, usually you use ZZ(123), which can run into problems if you end up dividing two ints.

I don't think it is so bad to get rid of all internal functions, but I can see some benefits to keeping some of them. However, the more trivial ones would be better to absorb as inline: psi_local, psi, beta_psi_f, beta_sigma_f, rho, beta_rho_f, rhopm, beta_rhopm_f, c3, c4, c0, omega. Many of these can be written as a prod and you could avoid refactoring chi.conductor() in a few places. (Also, it is better to use x -= y instead of x = -1 * y as a micro-optimization.) You should also always return a Sage Integer (in particular, line 896 to return ZZ.zero()).

[eross156]

Ok, thanks for the feedback. I will plan to add in these changes on Monday or Tuesday (I'm currently travelling as well).