Let Φ \Phi be a reduced root system of rank r r . A Weyl group multiple Dirichlet series for Φ \Phi is a Dirichlet series in r r complex variables s 1 , … , s r s_1,\dots ,s_r , initially converging for R e ( s i ) \mathrm {Re}(s_i) sufficiently large, that has meromorphic continuation to C r {\mathbb C}^r and satisfies functional equations under the transformations of C r {\mathbb C}^r corresponding to the Weyl group of Φ \Phi . A heuristic definition of such a series was given by Brubaker, Bump, Chinta, Friedberg, and Hoffstein, and they have been investigated in certain special cases by others. In this paper we generalize results by Chinta and Gunnells to construct Weyl group multiple Dirichlet series by a uniform method and show in all cases that they have the expected properties.

We construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet series in greater than two variables having the properties predicted in [2].1For convenience we assume that O ∈ E and m O = 1. Let J (S) be the group of fractional ideals of O coprime to S f . Let I, J ∈ J (S) be coprime. Write I = (m)EG 2 with E ∈ E, m ∈ K × , m ≡ 1 mod C, and G ∈ J (S) such that (G, J) = 1. Then, following [15], the quadratic residue symbol mm E J is defined, and if I = (m ′ )E ′ G ′ 2 is another such decomposition, then E ′ = E and m ′ m E J = mm E J . In view of this define the quadratic residue symbol I J to be mm E J . For I = I 0 I 2 1with I 0 squarefree, we denote by χ I the character χ I (J) = χ I 0 (J) = I 0 J .

In a previous paper [1], we de®ned the space of toric forms Tl, and showed that it is a ®nitely generated subring of the holomorphic modular forms of integral weight on the congruence group q 1 l. In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L f Y 1 Q 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols. First, in O2we recall results about the Manin symbols. We discuss various homology groups associated to the modular curve in terms of modular symbols, and

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Let Γ be a torsion-free finite-index subgroup of SL n (Z) or GL n (Z), and let ν be the cohomological dimension of Γ. We present an algorithm to compute the eigenvalues of the Hecke operators on H ν−1 (Γ; Z), for n = 2, 3, and 4. In addition, we describe a modification of the modular symbol algorithm of Ash-Rudolph [10] for computing Hecke eigenvalues on H ν (Γ; Z).