Fish Oil Supplements & Coumadin

After reading the expanding literature on heart health and fish oil supplements, I would like to begin taking this supplement on a regular basis (about 1 gram/day) .  I have had two surgeries for a congenital heart defect and now have 2 artificial heart valves.  I am taking coumadin (target INR 2.5 - 3.5).  I live overseas and have very limited access to fresh/frozen fish.  To regulate my INR, I use a portable tester that works well.  I'm wondering if the blood thinning effects of fish oil (if any) will be measured by my INR testing machine.  In other words, will the total blood thinning effects of both coumadin and the fish oil suppliments be measured by the INR?