Hecke Subalgebras and Local Newforms for the Metaplectic Double Cover of $\SL_2(\mathbb Q_p)$

We determine an explicit compact Hecke subalgebra for the metaplectic double cover of $\SL_2(\mathbb Q_p)$ at the congruence subgroup $K_0(p^n)$, for odd $p$, and use it to study local newforms of prescribed quadratic type. We describe the supporting double cosets, generators, relations, characters, and corresponding $\ov K$-types, and compute the action of the resulting Hecke operators on $(K_0(p^n),η)$-isotypic vectors in principal series, Weil, Steinberg, and supercuspidal representations. Thus the conductor relevant throughout is the $η$-conductor, rather than the minimum over all characters. In the supercuspidal case, the metaplectic calculation is reduced to the corresponding linear strongly cuspidal type, making explicit the distinction between unramified and ramified $L$-packets. We also compare the operator $W_{m-1}$ with Ishimoto's local realization of Ueda's twisting operator: after fixed-level compression and a lifted $\GL_2$-conjugation, the two actions agree up to an explicit scalar and a parity-dependent change of type. These results provide the local odd-prime counterpart to the Hecke-algebra methods used in the theory of half-integral-weight newforms and minus spaces.