  We investigate the large weight (k --> oo) limiting statistics for the low
lying zeros of a GL(4) and a GL(6) family of L-functions, {L(s,phi x f): f in
H_k(1)} and {L(s,phi times sym^2 f): f in H_k(1)}; here phi is a fixed even
Hecke-Maass cusp form and H_k(1) is a Hecke eigenbasis for the space H_k(1) of
holomorphic cusp forms of weight k for the full modular group. Katz and Sarnak
conjecture that the behavior of zeros near the central point should be well
modeled by the behavior of eigenvalues near 1 of a classical compact group. By
studying the 1- and 2-level densities, we find evidence of underlying
symplectic and SO(even) symmetry, respectively. This should be contrasted with
previous results of Iwaniec-Luo-Sarnak for the families {L(s,f): f in H_k(1)}
and {L(s,sym^2f): f in H_k(1)}, where they find evidence of orthogonal and
symplectic symmetry, respectively. The present examples suggest a relation
between the symmetry type of a family and that of its twistings, which will be
further studied in a subsequent paper. Both the GL(4) and the GL(6) families
above have all even functional equations, and neither is naturally split from
an orthogonal family. A folklore conjecture states that such families must be
symplectic, which is true for the first family but false for the second. Thus
the theory of low lying zeros is more than just a theory of signs of functional
equations. An analysis of these families suggest that it is the second moment
of the Satake parameters that determines the symmetry group.
