Here is the full list of all 16 Lucky Gem achievements.
Earn 100 coins
Earn 500 coins
Earn 1000 coins
Earn 10000 coins
Earn 100000 coins
Synthesize Lv.3 gems
Synthesize Lv.6 gems
Synthesize Lv.9 gems
Synthesize Lv.12 gems
Synthesize Lv.15 gems
Synthesize Lv.18 gems
Unlock 6 pictures
Unlock 12 pictures
Unlock 18 pictures
Unlock 24 pictures
Unlock 30 pictures