Dear Imin,

   How have you been doing? 

Great, ang I hope you are as well.

   Are you finished yet or already finished your thesis? 

I will finish this year.  I'm writing up my thesis right now and applying
for jobs.  If you have any suggestions as to where I should apply please
let me know. 

   I had a few questions about your programs to compute modular forms (by the
   way, I found your tables very useful for checking some results about
   congruences, thanks for posting them!) as they came up when I wanted to

Great!

   check some results about congruences. My question is: how high a conductor
   can you compute modular elliptic curves? Cremona is probably up to 6000 by
   now, but the problem is that the elliptic curves I'm looking for seem to
   have big conductor.

It depends on exactly what you want to do.  If you want a list of
Weierstrass equations for elliptic curves, like in Cremona, I'm
think John's programs can easily beat mine.   Nonetheless, I have
programs for computing the action of Hecke operators on spaces
of modular forms.  If you can work mod p they can be quite efficient,
for example:

> time H:=ModularSymbolsPlus(2,eps(6000,5));
Creating M_2(Gamma_1(6000),eps;F_5)...
Step I.   Manin symbols list.
122.27  seconds.
Step II.  2-term relations.
29.03  seconds.
Step III. 3-term relations.
# relations =  4800
vector space reduction
65.71  seconds.
Created M in 217.279 seconds.
Time: 217.389           
> time T:=Tn(H,2);
Time: 45.409
> Ncols(T);
1208
> time K:=Kernel(T);
Time: 1.620
> Dimension(K);
600                   

 
   I'm enclosing the slides for a talk I recently gave which explains
   a bit what I'm interested in. Basically, I'd like to find the newform
   attached to an elliptic curve whose mod 3,5,7,11 representations have
   image lying in the normalizer of a Cartan subgroup (this can be
   checked by seeing if the j-invariant has a certain form), and check that
   this newform is congruent mod 3,5,7,11 respectively to a CM-form at that
   level. For 3,5 there are a few examples in Cremona's range, but otherwise
   they are out of range.

Do you already have lists of Weierstrass equations for elliptic curves 
whose mod 3,5,7,11 representations have image lying in the normalizer 
of a Cartan subgroup?  (I.e., is there an X_ns(7)/Q that you understand?)
If so, then we start with an elliptic curve E/Q.  We know its modular
form, since ap = p+1 - #(Fp). The level is the conductor N=N_E.  So you
must then look at the modular forms of that level (using my program, say)
and check for appropriate congruences.  If this is the structure of the
problem then I might be able to help you.

   If you're interested, I'd be happy to send you my paper which gives the
   theoretical results. 

Yes -- thanks.  I'm happy to see applications where knowing
information about forms corresponding to higher dimensional abelian
varieties helps us to understand elliptic curves. 

   how one can get it running on an IRIX system if at all possible? 

I have just (during the last week) implemented many of my programs
in the newest version 2.5 of the computer algebra system Magma 
         (http://www.maths.usyd.edu.au:8000/u/magma/)
Magma is available for IRIX, Linux, Sun, DEC, IBM RS600, and HP-UX.
Unfortunately it is not free as they have a team of paid programmers.
I've only just implemented the modular symbols algorithms under 
magma, so my implementation may still be a bit rough.  Nonetheless,
I recommend it over the current C++ implementation.  It might be
a lot of work to get my c++ program running on your IRIX machine
but Magma already runs on IRIX. 

  Best,
   William