I live with my mother and she smokes; how bad is this for my newborn?

Risky for your Baby. Babies that are exposed to second hand smoke double their risk for sudden infant death syndrome. They tend to be colicky, more prone to painful ear infections, have respiratory problems and may develop asthma, never mind the toxins they are exposed to. No one should smoke in a house with a baby. Ever.
It could be very bad. Babies exposed to second-hand smoke are at higher risk of developing ear infections, lower respiratory problems (eg. Asthma and pneumonia), sudden infant death syndrome. If it is not possible for her to stop smoking, she should at least be in the habit of only smoking outside and changing her clothes when she returns. See more at http://no-smoke.Org/document.Php?Id=212.
Bad. Period. Risks are tremendous for newborns including sids (sudden infant death syndrome), also known as "crib death." increased risks for ear infections, upper respiratory infections, asthma, death from asthma, gerd (reflux), and many more are greatly increased with even one smoker in the house (or the car, even with the windows down!) ask grandma to smoke outside so that baby can quit being a smoker.
Very Bad. Second hand smoke is one of the most serious environmental hazards for babies and young children. Second hand smoke causes respiratory problems and infections. Educate yourself about the dangers, do what you can to minimize your baby's exposure, and help your mother with efforts to quit smoking.