  The electromagnetic coupling $\alpha$ and the electroweak mixing angle
$\theta_{\mathrm{W}}$ are parameters of the Standard Model (SM) that enter
precision SM tests and play a fundamental r\^ole in beyond SM physics searches.
Their values are energy dependent, and non-perturbative hadronic contributions
are the main source of uncertainty to the theoretical knowledge of the running
with energy. We present a lattice study of the leading hadronic contribution to
the running of $\alpha$ and $\sin^2\theta_{\mathrm{W}}$. The former is related
to the hadronic vacuum polarization (HVP) function of electromagnetic currents,
and the latter to the HVP mixing of the electromagnetic current with the vector
part of the weak neutral currents. We use the time-momentum representation
(TMR) method to compute the HVP on the lattice, estimating both connected and
disconnected contributions on $N_{\mathrm{f}}=2+1$ non-perturbatively
$O(a)$-improved Wilson fermions ensembles from the Coordinated Lattice
Simulations (CLS) initiative. The use of different lattice spacings and quark
masses allows us to reliably extrapolate the results to the physical point.
