First examine the fabric, look out for pearlescent sheen (the color of the shine will change slightly depending on how you hold it against light).
Consider the weight and price of the saree. A pure silk saree is usually ten times more expensive than artificial/synthetic ones.
Spun silk costs around ₹ 4500/- per kilogram. So, if your saree costs less than that, you need to check the authenticity.
Also, you can perform the burn test on the saree but remember to extract a loose thread in such a manner so that it does not ruin it.