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Details regarding the construction of the database of classical modular forms can be found in the paper Computing Classical Modular Forms [arXiv:2002.04717] by Alex J. Best, Jonathan Bober, Andrew R. Booker, Edgar Costa, John Cremona, Maarten Derickx, David Lowry-Duda, Min Lee, David Roe, Andrew V. Sutherland, and John Voight. In particular, the classical modular forms data in the LMFDB comes from three sources:

For newforms of weight $k=1$ the projective fields cut out by the kernel of the projective Galois representation associated to the newform come from four sources:

Additional acknowledgments

David Lowry-Duda provided the graphical presentation of classical modular forms, described in [arXiv:2002.05234].