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Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms. (English) Zbl 1158.11023
Author’s summary: The author proposes a generalization of the algorithm he developed previously [Exp. Math. 14, No. 4, 457–466 (2005; Zbl 1152.11328)]. Along the way, he also develops a theory of quaternionic \(M\)-symbols whose definition bears some resemblance to the classical \(M\)-symbols, except for their combinatorial nature. The theory gives a more efficient way to compute Hilbert modular forms over totally real number fields, especially quadratic fields, and he illustrates it with several examples. Namely, he computes all the newforms of prime levels of norm less than 100 over the quadratic fields \(\mathbb{Q}(\sqrt{29})\) and \(\mathbb{Q}(\sqrt{37})\), and whose Fourier coefficients are rational or are defined over a quadratic field.

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11Y16 Number-theoretic algorithms; complexity
Full Text: DOI
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