What's the best record toward Selberg's eigenvalue conjecture:
a Maass form on $\Gamma\_0(N)$ has eigenvalue greater than or equal to 1/4?



Best record toward Selberg's eigenvalue conjecture?
\[\frac{1}{4} - \left(\frac{7}{64}\right)^2 = \frac{975}{4096} \approx 0.238037\ldots\]
as far as I know. Established in an appendix by Kim and Sarnak to Kim's "[Functoriality for the exterior square of $GL\_{4}$ and the symmetric fourth of $GL\_{2}$](http://www.ams.org/mathscinet-getitem?mr=1937203)", Journal of the American Math. Society, 16 (2003). 


The above is "over the rationals" (which was/is my reading of the question), in comments work on the problem for general number fields got mentioned by [Kim and Shaidi, Cuspidality of symmetric powers with applications. Duke Math. J. 112 (2002)](http://www.ams.org/mathscinet-getitem?mr=1890650) and the more recent work of [Blomer and Brumley, On the Ramanujan conjecture over number fields. Annals of Math. 174 (2011)](http://arxiv.org/abs/1003.0559) which among others establishes the above for general number fields.


For generalizations in a different direction let me also mention [Bourgain, Gamburd, Sarnak, Generalization of Selberg's 3/16 Theorem and Affine Sieve, Acta Math. 207 (2011)](http://arxiv.org/abs/0912.5021) establishing 'a generalization of Selberg's theorem for infinite index "congruence" subgroups of SL(2,Z).'


