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2r+l abelian algebraic number arithmetic assume Atkin automorphic forms Brandt Matrix character common eigenfunction congruence relation conjecture conjugate Corollary corresponding cusp forms defined denote Dirichlet series divisor Eichler Eisenstein series element elliptic curve elliptic modular Euler product exists factor finite fixed forms of weight formula Fourier coefficients Fourier expansion Frobenius function f function field functional equation Galois group Hecke differential Hecke operators Hilbert modular forms integer isomorphic Jacquet Lecture Notes Lemma Let F mapping matrix maximal order modular functions modulo multiplicative multiplier system notation Notes in Math number field obtain p-adic L-functions positive integer proof Proposition proved quadratic extension quadratic field quaternion algebra quotient ramified rational number real number representation of G residue field resp respect result satisfies Serre Shimura subfield subgroup subspace supersingular points Suppose Theorem theta series totally real trace transformation trivial unramified zeta function