It used to be the case, 10 or so years ago, that William Stein maintained a website with extensive tables of \(q\)-expansions of modular forms, computed using magma. However, as the edifice of civilization begins to crumble, this website no longer works, and it no longer seems possible to easily access q-expansions online. The only current online resource I can find appears to be the modular forms database. Unfortunately, it appears that the designers of that website have invested more time into aesthetics than usability. First, some minor grumbles. The choice of how to represent Dirichlet characters seems somewhat cumbersome, at best. This is somewhat excusable, because there isn’t any obvious elegant solution, although the best choice will surely be one which is easier for the viewer to use. It’s also a little unclear, certainly in comparison to William’s old site, what range of levels and weights have been computed. More seriously, after spending five minutes chasing down the right Dirichlet character for a particular space of modular forms I wanted to compute, I clicked on the appropriate link only for it to misdirect me to a page with the same level and a different character. Indeed, this character even had the wrong sign, and so all the web page did was to unhelpfully inform me that the space of forms was zero. I then started clicking randomly on other characters in the hope that they would send me the the web page with the character I actually wanted, but no such luck. Another very annoying “feature” of the website is the format in which the answers are returned: they are given in (compiled) LaTeX! This is a terrible design decision. Given that I (and possibly other users) actually want to do something with the answer (for example, compute the reduction modulo a prime using pari), a pretty formula is completely useless, because it is impossible to cut and paste it in any useful way. At one point I did find a way to ask for the answer to be given in text format, but looking at the page again now I can’t even remember what I did. Even that was not so useful, because it didn’t actually work on my particular example and instead and caused the website to crash. In fact, the only way I was ever able to extract the answer in some useful way was to tell Safari to give me the source code of the web page. I should note that there is a “feedback” button at the top right of the website. On both occasions when these problems occurred, I dutifully submitted my questions/requests with a detailed explanation of what I did, but neither time did I get any response, nor any indication that those messages were even read. In fact, the website does not seem to contain information any about who maintains it (beyond that it is funded by the NSF), so I can’t even email someone directly to complain. So here I am complaining, in the hope that one of my readers has more influence with the designers of this web site than I do.
Update: This post has already been successful — various people have already contacted me to say they are working to address my concerns. Of course, this is what I expected, since the people I know who do these computations are genuinely interested in their data being useful to others…
Update II: Instead of recommending how to deal with characters, let me just mention what I often end up wanting to compute, namely, a space of weight one modular forms of level \(\Gamma_1(N)\) and character \(\chi\) in characteristic \(p.\) In practice, this means that I compute in weight two, level \(\Gamma_1(Np)\), and character \(\chi \cdot \epsilon\), where \(\epsilon\) is the mod-p cyclotomic factor. This ultimately requires making a number of choices of embeddings \(\overline{\mathbf{Q}} \hookrightarrow \overline{\mathbf{Q}}_p\) and \(\overline{\mathbf{Q}} \hookrightarrow \mathbf{C}\) which have to be compatible in some way but ultimately are irrelevant. So even though (for example) if \(\chi\) is a quadratic character (say) then \(\chi \epsilon\) is easy to describe in words, it becomes a little bit of a mess to get the LMFDB to give me the information I need…
http://www.lmfdb.org/editorial-board
Just because you know how to use the interwebs…
Just for future readers: The footer points to the source code, which is here: https://github.com/LMFDB/lmfdb/
I’ve helped creating those Dirichlet character pages. IIRC, the main issue is that at about 3 individuals brought up at least 4 completely different ideas how this should be done — where each idea was essentially just a vague description. No surprise this doesn’t work out as it should.
Could you give us a clear and detailed description how you would do this?
I plan to — I will update this comment when I have more time.
(I ultimately added a few more remarks in the main body of the blog post.)
I clicked on “Modular Forms – Classical, GL(2)/Q” on the left bar, then decided to browse and entered level 25, weight 4, on Gamma_1. This took me to a page entitled “Newforms of weight 4 for Γ1(25)”… I clicked on the link next to character “χ25(24,⋅)” (which indicated the space had dimension 4), and
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/25/4/5/
The page you are looking for has produced a server error.
Please keep your faith in mathematics!
Number of pages successfully viewed: 3
Number of attempted views that produced a server error: 1
Overall completion rate: 75%
Mean time to failure (1 sample): 4 views
I must admit I do have more faith in mathematics than in the LMFDB.
OK, I tried again.
Clicked on “Elliptic Curves/Q” on the left, then browsed by conductor by clicking on the “1-99” button, scrolled down to “37.a1”, clicked on this to be on the page “Elliptic Curve 37.a1 (Cremona label 37a1)”… Upon clicking on the right side Related objects “Modular form 37.2a” and
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/37/2/0/a/
Cuspidal newform a of weight 2 for Γ0(37) with trivial character
We are very sorry. The sought function could not be found in the database.
Number of pages successfully viewed: 4 (again counting the home page)
Number of attempted views that could not be found: 1
Overall completion rate: 80%
Mean time to failure/sorry (2 samples): 4.5 views
So it looks like the LMFDB is not yet up to say Antwerp-IV standards yet.
Third and final attempt…
Clicked on “L-functions Degree 2” (the “2” link) on the left. Goes to a browseable page. The most interesting thing to me was a K3 surface labelled “7.21458” (the lowest zero)… and clicking on this button gave
http://www.lmfdb.org/L/ModularForm/GL2/Q/holomorphic/7/3/3/a/0/
Sorry, there has been a problem: (‘unexpected EOF while parsing’, (”, 1, 17, ‘Notabletocompute.’)). Please report it here.
Oh well, maybe I can get to this by clicking on “holomorphic” from the same line of the table…
http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/7/3/3/a/
The page you are looking for has produced a server error.
Please keep your faith in mathematics!
Number of pages successfully viewed: 2 (again counting the home page)
Number of attempted views that failed: 2
Overall completion rate: 50%
Mean time to first failure (3 samples): 4 attempted views
I don’t think the word “annoying” is sufficiently uncharitable.
I hadn’t seen this blog post until just now. Anyway, here is an example of a public worksheet illustrating computing S_4(25,eps) for your above example in SageMathCloud: https://cloud.sagemath.com/projects/4a5f0542-5873-4eed-a85c-a18c706e8bcd/files/support/2015-09-02-054941-modular-forms.sagews
Sorry my old contribution to civilization crumbled. I was diverted first in about 2004 for a “while” to write Sage, so that I could create better tables without relying on closed source software, and next for a while longer by SageMathCloud, so that I would have a better way to collaboratively and interactively do such computations. Maybe I’ll do something useful when I get back from these diversions… 🙂
Thanks, a useful link! (Plus, I will know where to find it as well.)
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