  We give an asymptotic formula with power saving error term for the twisted
first moment of symmetric square L-functions on GL(3) in the level aspect. As
applications, we obtain non-vanishing results as well as lower bounds of the
expected order of magnitude for all even moments, supporting the random matrix
model for a unitary ensemble. Besides the GL(3) Kuznetsov formula, the
ingredients include detailed local computations at ramified places, including
root numbers and orthonormalization of oldforms and Eisenstein series.
