Date: Fri, 24 Jul 1998 13:46:31 -0700 (PDT)
From: "Robert F. Coleman" <coleman@math.berkeley.edu>
To: was@bmw.autobahn.org
Subject: Great!


William,

        By new in the following, I just meant new at 2, i.e. not coming
from X_0(49).  The dimension of these is one less than the number of
supersingular points on X_0(49) mod 2.  Now I know the eigenvalues in
question are roots of unity.  I guess they're \pm 1.  The number of 1's
should be the number of SS points defined over F_2 and the
number of -1's should be the number of SS points defined over F_{4}.
I'm not sure about this.

"In your lettter to Jeremy you said, "there are 5 new forms on
X_0(49*2) which gives (1-T)^5".  I'm not sure exactly what you meant
(the space of classical weight 2 cuspidal new forms at level 98 has   
dimension 3) but it gave me the idea to look there."


"In your letter to Jeremy you said that the 8 slope 0 Eisenstein
series have twins which give 8 slope 1 factors of R_1(T).  Also, it
seems that alpha gives two factors. What are the other 4?  Are they
somehow twins of the forms giving rise to the factor (1-T)^2 *(1 +
T)^3, or is that wrong?"

Actually only 7 of the 8 slope 0 Eisenstein series have twins.  It is
true that \alp gives two factors.  Thus there remain 5 mystery factors
which correspond to non-classical overconvergent forms.  I don't know much
about them.


"What is the next step?"


Two possibilities occur to me.  a) Prove my conjecture on the
valuations of the coefficients.  For this, one could use the ideas in
the appendix fo my Banach space paper. b)  Try to determine the
q-expansion of a non-overconvergent formk in the \alp isotypical
component.

As far as your other general questions go: The slope \alp subspace is
stable under Hecke. Not much is known about higher slopes.

        Robert