This is the 9065th most frequent Japanese word.
渋く
"渋く" means "subtly cool, refined, or understated in a mature or tasteful way."
Here, '渋く' is used to describe someone's action carried out with a mature, refined, or understated demeanor.
彼は渋くコーヒーを飲んだ。
He drank coffee in a refined manner.
In this case, '渋く' modifies the process of dying fabric, conveying a subtle or tasteful quality.
渋く染め上げた生地は高級感がある。
The fabric dyed in a subdued tone has a sophisticated feel.
'渋く' in this instance refers to the movie's theme being mature, deep, and understated in an appealing way.