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If no father listed on birth certificate then the child would have been illegitimate. Sometimes a father might be listed in a baptism record but 1940 too recent to be online (lots of churches in Coleraine- Catholic, Church of Ireland, Presbyterian, Methodist, Baptist, Congregationalist- and more outside Coleraine). Remember that 1945 was wartime and there were soldiers stationed in the area, local men going off to fight, people moving from Belfast to escape bombing, etc. so lots going on in the area.
This topic which mentions New Row, although a good bit earlier, might be of interest-
https://www.rootschat.com/forum/index.php?topic=794540.0As you can see from these photos New Row West has changed a bit since 1940s-
https://www.geograph.ie/photo/2886347https://www.geograph.ie/photo/2873700If you have a street number there might be a listing in local directories although 'New Row West' might be as much detail as you'll get. It's possible relatives lived in the house where your father was born and it might be possible to get more information.
Once you've made 2 or 3 posts you'll be able to exchange information by PM (Personal Message). If you need help with details you'd rather not post online get in touch and I'll try to help.
Added-if you put 'new+row+west' into search box for 'townland or street' in 1911 census you'll see all 21 people living there on 2 Apr.1911-
http://www.census.nationalarchives.ie/search/