def ith_magical_component(omega, zvee, i):
    '''Given a form omega on AS cover and normal basis element zmag, find the decomposition
    sum_g g(zmag) omega_g and return omega_g, where g is the ith element of the group.'''
    AS = omega.curve
    p = AS.characteristic
    group_elts = [(j1, j2) for j1 in range(p) for j2 in range(p)]
    z_vee_fct = zvee.group_action(group_elts[i]).function
    new_form = as_form(AS, z_vee_fct*omega.form)
    return new_form.trace2()