Cuspforms of level 11048

[B Kim]

Dear all,

Me and my colleague have been trying to find cuspforms of weight 2 and level 11048 by Sagemath, which only produced errors.

Since LMFDB has an entry for weight 2 and level 11048 (https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/?level=11048&weight=2&char_order=1), I guess it is computable. But the LMFDB entry doesn't have q-expansions (and in fact, we need fairly long q-expansions).

What software does LMFDB use to find them? Does LMFDB have the original data? (If long q-expansions are available, all the better.)

Best regards,

BD

[Andrew Sutherland]

We used Pari/GP (which uses the trace formula) and Magma (which uses modular symbols) to compute most of the modular form data in the LMFDB (the implementations in both Pari/GP and Magma are more capable (in different ways) than the implementation in Sage).  For weight 2 and trivial character this includes levels up to 10000, as explained in https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Completeness and https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Source.  You can read more about the methods we used in the paper https://arxiv.org/abs/2002.04717.

For levels beyond 10000 the data we have was computed using two different methods, one of which is applicable only to prime levels (see https://doi.org/10.1007/s40993-022-00392-z), while the other uses a heavily optimized implementation of modular symbols and Atkin-Lehner operators (see https://dspace.mit.edu/handle/1721.1/156828) that can decompose the space into newform subspaces much more quickly than magma, and it can be used to compute trace forms, but it does not attempt to compute q-expansions.

Level 11048 with trivial character is still in the range where you should be able to compute q-expansions using Magma's modular symbol implementation (which is what we used in the range 4000-10000 for trivial character).  It is possible that Pari/GP can also handle this level, but I have less experience with Pari/GP in this range (my recollection is that Magma is faster here, but as you can see from Tables 7.1.1 and Tables 7.1.2 of https://arxiv.org/abs/2002.04717, each implementation has its strengths, and it depends on the shape of the level, which isn't really visible in these tables).

But you will need to be patient, as it will take a substantial amount of time to decompose the space (once the space is decomposed computing q-expansions is pretty quick).   You will also want to run this on a machine that has plenty of memory.

You can find the relevant Magma documention here: https://magma.maths.usyd.edu.au/magma/handbook/modular_symbols, and the Pari/GP documentation here https://pari.math.u-bordeaux.fr/dochtml/html/Modular_forms.html.

[B Kim]

Thank you. How much memory? I have access to cluster computing. The system administrator said it's probably equivalent to 50-100 desktop computers.

[Andrew Sutherland]

My guess is 64GB would be plenty, but I would try to have more available if you can, there is nothing worse than having a computation run for days and then crash in the last stage because it runs out of memory.  Unfortunately this computation is not easy to parallelize, so the cluster won't really help in terms of speed (each individual core will be slower than the cores on your laptop), but it should have plenty of memory.

[B Kim]

I see. Thanks a lot.

BD