Ariel Pacetti on Wed, 28 Mar 2012 16:52:39 +0200

 forqfvec

Dear pari developers,

sometime ago Bill wrote a routime called forqfvec which is a mix of qfminim and somehow forvec (see http://pari.math.u-bordeaux.fr/archives/pari-dev-1005/msg00003.html).

The advantage of such routine is that instead of giving the whole output of qfminim (which might blow up the memory) it lets you manage each vector at a time and then discard it. The first application of it is the routine qfrep writen by Karim.

In such routine, one takes each vector of qfrep and compute its norm, and then puts a 1 in the right place of a vector. This gives the q-expansion of modular forms (of integral weight or half integral weight) which is quite useful. I was asked by Bill to make some lobby on this routine, so let me just mention some other applications of it so as to consider including it in the future releases.

1) Say you want to compute theta series of quadratic forms. Then the routine qfrep does the job. But if you want to compute the form with a character for example, then you need the forqfvec routine. I used such routine in the paper "Computing central values of twisted L-series, the case of composite levels" with Gonzalo Tornaría. At the time, Gonzalo implemented the particular routine in C.

2) Say you want to compute modular forms (from quaternion algebras) of weight greater than 2. Then what you do is to add an harmonic polynomial to the theta function, and evaluate it in the elements for the theta series. Again, forqfvec will do the job (you can check Eichler's paper on the basis problem to see how this works).

3) Say you want to compute Hilbert modular forms (as in Lassina Dembele's works). Then you have forms over a totaly real field and need to evaluate the elements at some representation (if the class number is greater than one). Again, forqfvec does the job. Furthermore, I wrote a script for a particular case of a HMF over Q[\sqrt{5}] to compute many Fourier coefficients using forqfvec (and was the first time I asked Bill such routine).

There should be many other applications relating modular forms that I am missing, so I believe having it in the official release might be quite useful.

Cheers,

Ariel