GL2xGSp2 L-values and Hecke eigenvalue congruences

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as GSp2(A), SO(4, 3)(A) and SO(5, 4)(A), where the prime modulus should, for various reasons, appear in the algebraic part of a critical “tensor-product” L-value associated to cuspidal automorphic representations of GL2(A) and GSp2 (A). Using special techniques for evaluating L-functions with few known coefficients, we compute sufficiently good approximations to detect the anticipated prime divisors.