Under perfect sky conditions the Moon doesn't need to be illuminated at all, and you might experience a solar eclipse. In other months it is approximately nearly a day after new moon.
Even during new moon the Moon's dark side will be illuminated by the Earth shine, thus it is not totally dark. So even without an eclipse one can in principle also spot the new Moon in the vicinity of the Sun. Practically it's a matter of contrast, and that makes the problem surprisingly difficult.
https://astro.ukho.gov.uk/moonwatch/nextnewmoon.html gives a good overview over where and when you can spot the new moon with which optical aids (and if any are needed). According to that the earliest you can see it with the naked eye is around 15..23h after new moon and 12..19h with a telescope. The exact duration between new moon and first visibility varies, depending on where the Earth and Moon are in their respective orbits, also due to the the inclination, thus the height of the Moon in the sky - which in turn affects visibility and contrast:
Predicting the first sighting of the new crescent moon has been attempted since the time of the Babylonians and perhaps earlier still. These early methods relied on the age of the Moon and the time difference between moonset and sunset. More modern methods use the angular separation of the Sun and Moon, the altitude difference between the Sun and Moon, their relative azimuths and the width of the crescent. Some methods even try to model atmospheric effects to improve the prediction and to address the effect of relevant atmospheric phenomena.
They also give an extensive technical treatise on the problem describing their approach(es) in detail.
You can also visit this site to have it tell you for your location the first visibility.