The results in Section 1.1 strongly suggest that if
is a newform, then asymptotically 0% of
are in
(for
prime). If you actually look at the data,
a large proportion of
do lie in
for relatively small
;
as
gets larger the proportion (appears to) tend to 0
. In
contrast for every newform
of degree
that I (=Stein) investigated the proportion of
was very
very small.

Problem 1.2.1
Explain the above observations about the proportion of
that
lie in
. Gather further numerical data.