Moments of symmetric square L$L$‐functions on GL(3)${\rm GL}(3)$

Abstract We give an asymptotic formula with power saving error term for the twisted first moment of symmetric square ‐functions on in the level aspect. As applications, we obtain nonvanishing 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 Kuznetsov formula, the ingredients include detailed local computations at ramified places, including root numbers and orthonormalization of oldforms and Eisenstein series.