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**Sample text**

E. , if g corresponds to cp then (g, 0(2L,) (Ij)) (p, cp)2 _ (g" g`) (cp(yti) cp(yj)) holds. Proof. Let Mbe a divisor of N and g E S2+2v (M') and let a be a function from the set S of prime divisors of N/M' (whose cardinality we denote by w(M/M')) to {f1}"(M/M'). Then the function gE :_ >s'CS 91 fpES e(p)WP) in S2+2v (N) is an eigenfunction of the Atkin-Lehner involutions w p for the p I (N/M') with eigenvalues e(p). Similarly, fix a maximal order R C D and an Eichler order R = R(Ml, M2) C_ R, let M2 I M2 and let R(M1i M2) be the Eichler order of level M1M2 in D containing R and contained in R.

1) we are free to interchange the roles of ¢ and V (interchanging the roles of v2 and v3 at the same time), but f has to be the cusp form of largest weight. g. in case IB, IIB we should also consider the case where p does not divide the level of 0 or 0. It will be left to the reader to show by similar computations as above that in those cases the Euler factor will be the same (as should be more or less clear from an adelic point of view). 4. - The Functional equation The factors T,,. = T ,,(s) computed in the previous section are for p f N and for p I gcd(Nf, No, N,,) up to a shift in the argument and an elementary factor the known Euler factors of the triple product L-function L(f, ¢, z/), s) associated to f, 0, z&.

22) T. SATOH. - Some remarks on triple L -functions, Math. Ann. 276 (1987), 687-698. [23) H. SHIMIZU. - Theta series and automorphic forms on GL2, J. of the Math. Soc. of Japan 24 (1972), 638-683. [24) G. SHIMURA. Pure Appl. Math. 29 (1976), 783-804. [25) G. SHIMURA. - The arithmetic of differential operators on symmetric domains, Duke Math. J. 48 (1981), 813-843. S. Bicherer and R. Schulze-Pillot Fakultat fur Mathematik and Informatik Universitat Mannheim Seminargebaude AS D-68131 MANNHEIM Number Theory Paris 1993-94 Abelian extensions of complete discrete valuation fields Ivan B.