# HMF: simplify Hecke field polynomials #975

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opened this Issue Mar 31, 2016 · 3 comments

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### AurelPage commented Mar 31, 2016

 From /roadmap: Simplify presentation of Hecke field (adjoin all elements to get an order, or mimic classical modular forms by working with the dual).

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### edgarcosta commented May 14, 2017

 An example (for future reference): The Heck eigenvalue field of http://www.lmfdb.org/ModularForm/GL2/TotallyReal/2.2.5.1/holomorphic/2.2.5.1-331.1-a is http://www.lmfdb.org/NumberField/6.6.50840384.1 and we should be using the integral basis computed for 6.6.50840384.1 to represent the eigenvalues.

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### SamSchiavone commented Jul 20, 2017

 I wrote a script to express the Hecke eigenvalues for Hilbert modular forms in terms of an integral basis. Given a hmfs label, the script extracts the polynomial defining the field containing the Hecke eigenvalues, runs polredabs on it, then looks up the number field in the LMFDB, extracts the integral basis stored there, and re-expresses the Hecke eigenvalues in terms of this integral basis. How would we like to display this information on the pages for the Hilbert modular forms? Maybe write alpha_1, ..., alpha_n for the elements of the integral basis and write the eigenvalues as a linear combination of them? Currently the script outputs tuples of coefficients for each eigenvalue, the number field, the field generator, and the integral basis. Should we keep track of the number fields that we find that do not have an LMFDB page? Currently the script just outputs "No number field file found in LMFDB" when this happens.
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### edgarcosta commented Sep 15, 2017

 The pr #2197 makes some progress towards this. Perhaps we should write here, what are the next steps.