Timings for Tahajjud Prayer

Salaam, Actually I am a little confused I hope someone can assist me. I want to pray Tahajjud but I am confused about the timings. I usually pray Isha after I come home at around 12 Am and I usually go sleep at 2 or 2:30 am. Now I also have to wake up at 5 am for Fajr. So i want to ask can I pray Tahajjud at 2 am when I am about to sleep? Because if i go to sleep at 2 am then it won't be possible for me to wake up again at 3 am to pray Tahajjud. So can I pray Tahajjud at 2 am? Fajr time in my city starts at around 4:20 am.