Newforms of half-integral weight: the minus space of Sk+1/2(Gamma(0)(8M)): the minus space of S_k+1/2(Γ₀(8M))

We compute generators and relations for a certain 2-adic Hecke algebra of level 8 associated with the double cover of SL₂ and a 2-adic Hecke algebra of level 4 associated with PGL₂. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the 2-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight k + 1/2 and level 8M where M is odd and square-free. Using these operators and our previous results on half-integral weight forms of level 4M we define a subspace of the space of half-integral weight forms as a common -1 eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace, which we call the minus space, is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight 2k and level 4M. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.