In what specific ways does Hedda Gabler spearhead the movement for feminism or the new woman?
Spearhead? None. That would indicate it leads the way, and it did not.
Hedda Gabler does do some modest work at showing alternative ways for people to act. Hedda herself is a striking character, full of extroverted energies. She is intelligent and sexual, and works to shatter some social expectations about women. Likewise, her desire to shape a person's destiny fits with the new woman's desire for self-determination.
However, her value as a symbol of new woman is limited by her suicide, and, before that, by the emphasis on sexual/romantic control over men (rather than sexual desire for her sake). The suicide does show what happens to women of power when they aren't allowed to live those powers.
self centered she was a selfish and showy