By G. H. & Wright, E. M. Hardy

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For a q ∈ G L n (K )h we define Dq (z, s; k, χ , c), a function on (z, s) ∈ H × C, associated to DA (x, s) by the rule (see [30, p. 146]), ˆ s). Dq (x · i, s; k, χ , c) = jxk (i)DA (diag[q, q]x, We now introduce yet another Eisenstein series for which we have explicit information about their Fourier expansion. In particular we define the E A∗ (x, s) := E A (xηh−1 , s) and DA∗ (x, s) := DA (xηh−1 , s), and as before we write Dq∗ (z, s; k, χ , c) for the series associated to DA∗ (x, s). We now write the Fourier expansion of E A∗ (x, s) as, q σ qˆ E A∗ c(h, q, s)eA (hσ ), (2) ,s = 0 qˆ h∈S where q ∈ G L n (K )A and σ ∈ SA .

V ∩r R ∗ q −1 where ξ ∈ V ∩ r R ∗ q −1 such that ξ ∗ τ ξ = σ . For our later applications we now work out the functional equation with respect to 0 −1n . In particular we are interested in the the action of the element η = ηn = 1n 0 −1 ∗ theta series θA (x, ω) := θA (xηh , ω). We note that by Eq. 4 (6)] we have that λ(x) = i p |N F/Q (det (2τ −1 ))|n λ(y)eh (−2−1 T r K /F (tr (y ∗ τ x)))dy, η Vh where p = n 2 [F : Q] and dy is the Haar measure on Vh such that the volume of 2 Mn (r)h is |D K |−n /2 .

291(2) (1991) 16. : Rational isogenies of prime degree. Invent. Math. 44(2) (1978) (With an appendix by D. Goldfeld) 17. : Arithmetic Duality Theorems. Kea Books (2004) 18. : On anticyclotomic μ-invariants of modular forms. Compositio Math. 147(5) (2011) 19. : On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math. 100(2) (1990) 20. : Euler Systems (Hermann Weyl Lectures). In: Annals of Mathematics Studies, vol. 147. Princeton University Press (2000) 21. : Multiplicative reduction and the cyclotomic main conjecture for GL2 .