Snow cover is the key element to the coldest weather in winter. There are two physical ways that snow cover keeps the weather colder: radiative properties and thermodynamic properties. First of all, the whiteness of snow reflects most of the incoming sunlight back into space, keeping the air colder. Less obvious, but also important, is that the white snow cover is also more efficient than dark ground at emitting certain wavelengths of terrestrial radiation into space, also making the air in the lower atmosphere colder.
Thermodynamics is also important. Specifically, the heat required to melt snow uses up energy that would otherwise warm the air. When the weather manages to warm above 32 degrees, it tends to remain in the 30s as long as the snow is melting. Temperatures in the 40s or 50s this time of year occur exclusively when there is no snow on the ground.