 
Summary: Some calculations regarding torsion and component groups
Note: What I am calling observations below are really hunches based on part of the data; they
have to be tested on more data. Let f be a newform of weight 2 on 0(N), and let A = Af be
the quotient of J0(N) associated to f. For the moment, suppose N is squarefree, and let K be
the full cuspidal subgroup of J0(N), which is necessarily rational.
Observation 0.1. A(Q)tor K.
I suppose to start with, one can check if A(Q)tor divides K, say for elliptic curves, us
ing your table http://modular.ucsd.edu/Tables/cuspgroup/index.html. I would not be too
surprised if things do not work out at the primes 2, 3 and the primes dividing N.
When N is not squarefree, I guess the above may be true with K replaced by the rational part
of the full cuspidal subgroup (i.e., A(Q)tor K J0(N)(Q)). But I have not looked at the data.
Suppose now on that N is any integer (but at times one may need to assume N is squarefree).
The rational divisor (0)  () generates a finite subgroup of J0(N)(Q), which we denote C.
The image of C under the quotient map J0(N) A is a cyclic subgroup of A(Q)tor; we denote
this subgroup by CA and call it the cuspidal subgroup of A (note that this is not the subgroup
generated by the images of all the cuspidal divisors, but by the image of just (0)  ()). Let wp
denote the eigenvalue of the AtkinLehner involution Wp acting on f. The product of the Wp's
for pN is the Fricke involution WN , whose eigenvalue is denoted wN .
The following observations are based on Cremona's data and your table
http://modular.fas.harvard.edu/Tables/non zeroinf tor.txt.
