Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields

We study non-vanishing modulo primes of critical values of $L$-functions over imaginary quadratic fields twisted by Coates–Wiles characters. For a parallel weight two Hecke eigenform over an imaginary quadratic field, we prove that a positive proportion of Coates–Wiles characters have nonzero integral $L$-values modulo each prime in a positive-density set. The argument constructs automorphic symbols in parabolic homology with integral coefficients and expresses the critical values through their pairings with parabolic cohomology classes. A vertical family of additive averages of these pairings recovers Fourier coefficients and force the module generated by symbols to have full rank. We transfer this full-rank property to non-vanishing modulo primes. This extends the homological strategy of Kim–Sun from classical modular curves to arithmetic orbifolds, while addressing the unit obstructions specific to the imaginary quadratic setting.