Classical Modular Forms
People:
W. Stein -- optimizing computation of a_p; running computations and organizing results
- S. Butt -- running SAGE parallel tasks on TACC supercomputer
- C. Citro -- getting cyclotomic linear algebra into shape
- D. Roe -- plane text or database schema format for our data
- C. Pernet -- general exact linear algebra consulting
Status reports
Future plans
(Butt) Push a_p calculations for non-prime N up to 1000, p\leq 100,000 along with other "large" computations (e.g. higher weight, non-trivial character) on Lonestar. If you have large computations you want run on Lonestar, email me and we ought to be able to get your job going.
Goal
\Gamma_0(p), weight 2, for p\leq 10000
\Gamma_0(N), weight 2, for N\leq 3000
\Gamma_1(N), weight 2, for N\leq 256
fixes to make this run
\Gamma_0(N), weight 4, for N\leq 500
\Gamma_1(N), weight 4, for N\leq 100.
For each compute:
For primes p \leq 10,000 compute a_p exactly.
For N\leq 1000 prime, compute a_p exactly for p\leq 100,000.
Compute 30-digit numerical approximation for the a_n.
- The root number (Mark Watkins knows how to do this for nontrivial character). For trivial character this is minus the product of the Atkin-Lehner eigenvalues.
Data
Talks
- Cyclotomic linear
- Modular symbols
Supercomputing on Lonestar Slides
- Exact linear algebra: dense and sparse and *practical* discussion of: charpoly, kernel, echelon form, rational canonical form
Strategy
Optimize computing a_p; trac #3502.
- New subspace
- The command numerical_eigenforms is potentially very useful. It is fast but only double precision. It needs to be extended to nontrivial character, I think. Potentially not at all "proof = True".
sage: a = numerical_eigenforms(2000) sage: time b = a.systems_of_eigenvalues(10) CPU times: user 7.01 s, sys: 0.34 s, total: 7.35 s Wall time: 7.47 s sage: KeyboardInterrupt sage: b[0] [-3.81259043028 + 0.00283013715885*I, -4.0 - 2.63790725374e-14*I, 0.173299419789 - 0.000128642593614*I, -8.0 - 5.34372893157e-14*I]
Weight 1 Icosahedral Modular Forms
People: N. Ryan, N-P Skoruppa