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Thursday, October 31, 2019

Entry 42

Define dk=η(kτ) with the Dedekind eta function η(τ). Then for Level 8 (d34d2d28)44=(d21d4d2d28)2(d34d2d28)4+4=(d52d21d4d28)2
Or more simply as a4=ba+4=c
where (a,b,c) are the McKay-Thompson series of class 8E for Monster (A131125) and obeys (64abc+bca)+2a=abc+bca+acb
But one of addends is not a moonshine function so this is not one the 9 dependencies found by Conway et al. However, we also have a+16a=(d1d4)8+8a16a=(d2d4)12
 Adding the two together yields 2(d34d2d28)4=(d1d4)8+(d2d4)12+8
and this is one of the 9 dependencies found by Conway, Norton, and Atkins such that the moonshine functions span a linear space of 1729=163 dimensions.

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