Fun with Visualizing Modular Forms (in SAGE)

Problem 9.0.1   Implement a plot method in SAGE for modular forms. The following should thus make sense:

sage: E = EllipticCurve('37a')
sage: f = E.modular_form()
sage: plot(f)
...

Look at the code you get when you type E._plot_?? to see how to make a plot method for $f$ , so that plot(f) will work.

Regarding how $f$ should plot, probably there should be options, e.g., it could plot using a generic function for plotting a complex valued function, and all arguments to $f$ 's plot should get passed on to that generic function. But, you'll need to compute rigorous bounds on the number of terms of the $q$ -expansion that should be used, but allow the user to override them. See the paper by Kevin Grosvenor at

http://modular.math.washington.edu/papers/ell_graphs/ell_graphs.pdf

Problem 9.0.2   Somehow use the ray tracer that is included with SAGE to draw pretty pictures of modular functions. To get started, type Tachyon? in SAGE.

Problem 9.0.3   Helena Verrill wrote an awesome program for visualizing fundamental domains, both in Java and Magma (?):

     http://www.math.lsu.edu/~verrill/fundomain/

Write something like her program, but for SAGE. Here's an email she sent me (today!):

From: "Helena Verrill" <[email protected]>
To: [email protected]
Subject: Re: very fast GMP on OSX Macbook
Date: Tue, 1 Aug 2006 05:14:29 -0500
Hi William,
 
Thanks for sending the info about the upgrade. 
I guess I should start using the SAGE graphics - I ought
to get the fundamental domains pictures working in SAGE.

Have you met my students, Maya and Cristian? [...]
Thanks for the link to the wiki page.  It looks like a great
meeting.  
 
Best wishes,
Helena

Problem 9.0.4   There are efficient formulas for computing dimensions of spaces modular forms. Type dimension[tab] for the commands or see this page:

http://modular.math.washington.edu/sage/doc/html/ref/module-sage.modular.dims-doc.html

Draw graphs of some of these functions, as a function of $N$ , as a function of $k$ , and as a function of both.