Thursday, October 31, 2019
Entry 41
Define dk=η(kτ) with the Dedekind eta function η(τ). Then for Level 6 (d22d3d1d26)4−1=(d2d33d1d36)3(d22d3d1d26)4−9=(d51d3d2d56) Expressed as the triple of eta quotients (a,b,c) such that a−1=ba−9=c then (m,n)=(−1,−9) and abc=(d2d3d1d6)12bca=(d1d3d2d6)6acb=(d1d2d3d6)4 where (a,b,c) are the McKay-Thompson series of class 6E (A105559, A128633). They obey (64abc+bca)+2a=abc+bca+acb which is one of the 9 dependencies found by Conway, Norton, and Atkin such that the moonshine functions span a linear space of 172−9=163 dimensions. Similar identities involving only moonshine functions exist for levels (6,10,12,18,30).
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