Using molecular dynamics simulation of a standard coarse-grained polymer glass model we investigate by means of the stress-fluctuation formalism the shear modulus $μ$ as a function of temperature $T$ and sampling time $Δt$. While the ensemble-averaged modulus $μ(T)$ is found to decrease continuously for all $Δt$ sampled, its standard deviation $δμ(T)$ is non-monotonous with a striking peak at the glass transition. Confirming the effective time-translational invariance of our systems, $μ(Δt)$ can be understood using a weighted integral over the shear-stress relaxation modulus $G(t)$. While the crossover of $μ(T)$ gets sharper with increasing $Δt$, the peak of $δμ(T)$ becomes more singular. % It is thus elusive to predict the modulus of a single configuration at the glass transition.

5 pages, 6 figure, published at PRL