you could flower under the t5s but if I were you I would change over to the hps as autos are very responsive to different light spectrums.
I always advise that for the veg period you should white/blue light as this will give you nearly double the growth of a yellow/red light
For the flowering period red/yellow light will increase the size strength and overall quality of the bud.
Here is a nutrient schedule I use for autos. I know these are not the nutrients you are using but I thought I would post this to help you understand the lifecycle of an auto a little better.
Autos are very sensitive to nutrients so I always advise to start with half the normal/recommended dosage and then slowly increase the dosage until you find the right level of fertilisation for the strain.
Autos also have a need for root stimulation in most cases, this comes from the short veg cycle, so root stimulator in the early days will also massively increase your yield(in most cases)
I will be back with more soon I hope I am helping.