Sauk leader Black Hawk named for what bird? (link to picture)

Black Hawk's name in the Sauk language means literally "big black hawk," but it is far from clear what species that would be, with suggestions from the American kestrel to the American swallow-tailed kite. This article suggests that he was named for Harlan's hawk, which seems plausible.
https://sora.unm.edu/sites/default/files/journals/wilson/v044n02/p0078-p0087.pdf

However, as varied as the tailfeathers of Harlan's hawk are (no two are said to be alike), I can't identify it with the bird that Black Hawk's medicine bundle is made from in this painting, which is the bird he was named for:
https://americanart.si.edu/artwork/muk-tah-mish-o-kah-kaik-black-hawk-prominent-sac-chief-3925
perhaps better viewed here:
https://en.wikipedia.org/w/index.php?title=Life_of_Ma-ka-tai-me-she-kia-kiak:_Black_Hawk

What is that bird?

Grateful for your thoughts.

This message was edited Mar 18, 2021 8:13 PM