326 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD
For the next proposition we refer the reader to the discussion before and after
the last lemma in Section 1.7 of Clozel's article in these proceedings [Cl].
Proposition 4.10 (Kim-Shahidi [KS3]). a) Ramakrishnan's 90% estimate (Raj
for the density of primes where 1l"v is tempered can be improved to 34/35.
b) At each unramified v, let av = av+ f3v· Assume Wv = 1, where Wv is the
central character. Assume further that 7r satisfies the Ramanujan conjecture. Then
for each c > 0, there are sets r+ and r-of places of F with positive lower density
such that av > 2 cos (il) -c: for all v E T+ and av < -2 cos (in +c: for all v E T-.
Note that 2 cos (il) = 1.68... This improves Serre's earlier estimate (Se} towards
the Sato-Tate conjecture (Appendix to (Sh9}).