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tk +1 12,N,x (11 ) on the space of cusp forms S(k + 1/2, N, x) of half-integral weight under the assumption X2 = 1. The purpose of this calculation is to find a relation betw een these tra c e s a n d th o se o f t h e H ecke operators o f integral weight 2 k . W hen th e 2-order o f th e level N (= ord 2 (N )) is sm all, w e found certain relations between the traces, in [U]. In this paper, w e report relations for the remaining cases.

Supplement to the decomposition of the spaces of cusp forms of half-integral weight and trace formula of Hecke operators