forqfvec

[Ariel Pacetti <apacetti@math.utexas.edu>]

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