Multiplicity one theorems, S-version

It is well known by the strong multiplicity one that π is uniquely determined by the Satake parameter c(π, v) for almost all v. Also, it suffices for us to test only finitely many v. We proved some S-effective version of multiplicity one theorems. Roughly speaking, if π and π′ are not equivalent, then there is also a bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{N}(S)$$\end{document} which is some expression in terms of K, d and max(N(π), N(π′)), which are analytic conductor of π and π′, respectively (will be defined soon), such that there is a v ∉ S with πv ≌ π′v and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\mathfrak{p}_v < \mathfrak{N}$$\end{document}. We also proved S-effective multiplicity one for the Chebotarev Density Theorem, and for GL(1).