Sorry, bit late to the party

Myntboken agrees with the analysis above: the one with the "uptick" 7s is Stockholm and occurs only with the special Stockholm monogram, where the Fs cross much lower down their stems. The central horizontal crosspiece of each F is some distance above where the two cross, whereas on the Avesta pieces the crosspieces more or less coincide with where the Fs cross.
Price: these are from 2011 but show the relative difference between the mints. 7000 SEK for Stockholm vs 500 or 725 SEK for Avesta (all VF).
There are two Avesta varieties for 1737, hence the two prices above. The one illustrated above from
Sveriges mynt 1521-1977 has a smaller crown above the monogram and is the slightly cheaper variant. It would seem that the Avesta mint changed die during 1737 as the large crown variant (called "B" in Myntboken) was used 1730-1737 and the small crown variant ("C") used 1737-1744.
Ole, can you post the other side of your Avesta coin so we can attribute it to B or C? Or is it too worn to tell?