  We refine known dimension formulas for spaces of cusp forms of squarefree
level, determining the dimension of subspaces generated by newforms both with
prescribed global root numbers and with prescribed local signs of Atkin-Lehner
operators. This yields precise results on the distribution of signs of global
functional equations and sign patterns of Atkin-Lehner eigenvalues, refining
and generalizing earlier results of Iwaniec, Luo and Sarnak. In particular, we
exhibit a strict bias towards root number +1 and a phenomenon that sign
patterns are biased in the weight but perfectly equidistributed in the level.
Another consequence is lower bounds on the number of Galois orbits.
