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Mathematics of Computation

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Using Katsurada's determination of the Eisenstein series to compute Siegel eigenforms

Authors: Oliver D. King, Cris Poor, Jerry Shurman and David S. Yuen
Journal: Math. Comp. 87 (2018), 879-892
MSC (2010): Primary 11F46; Secondary 11F30
Published electronically: May 1, 2017
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Abstract: We compute Hecke eigenform bases of spaces of level one, degree three Siegel modular forms and $ 2$-Euler factors of the eigenforms through weight $ 22$. Our method uses the Fourier coefficients of Siegel Eisenstein series, which are fully known and computationally tractable by the work of H. Katsurada; we also use P. Garrett's decomposition of the pullback of the Eisenstein series through the Witt map. Our results support I. Miyawaki's conjectural lift, and they give examples of eigenforms that are congruence neighbors.

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Additional Information

Oliver D. King
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Neurology, University of Massachusetts Medical School, Worcester, Massachusetts 01655

Cris Poor
Affiliation: Department of Mathematics, Fordham University, Bronx, New York 10458

Jerry Shurman
Affiliation: Department of Mathematics, Reed College, Portland, Oregon 97202

David S. Yuen
Affiliation: Department of Mathematics and Computer Science, Lake Forest College, 555 N. Sheridan Road, Lake Forest, Illinois 60045
Address at time of publication: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, Hawaii 96822

Keywords: Eisenstein series, $F_p$ polynomial, Siegel eigenform
Received by editor(s): March 18, 2016
Received by editor(s) in revised form: August 31, 2016, and September 16, 2016
Published electronically: May 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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