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Thursday, October 31, 2019

Entry 51

Level 24, Part 2. As pointed out in Part 1, Level 24 is unusual since two triples (a,b,c) have the special property a1b1c1=a2b2c2 These also form linear dependencies but involve non-monster functions. But if we combine the two, plus some level 12 functions, then there is a relationship with only monster functions. Define dk=η(kτ) with the Dedekind eta function η(τ). Then (d2d12d4d6)(d3d8d1d24)2+1=d52d3d8d512(d1d4d6d24)3(d6d12d2d4)(d3d8d1d24)21=(d4d6)4d1d22d3d8d212d24 Expressed as the triple of eta quotients (a,b,c) such that a+1=ba1=c so (m,n)=(1,1). Then abc=(d22d3d8d212d1d24d26d24)4bca=(d2d4d6d12d1d3d8d24)2=T24Bacb=(d4d6d2d12)6 where (a,b,c) are the McKay-Thompson series of class 24I (A138688). They obey bca=a1a (4abc+bca)+2a=abc+bca+acb This is not one of the 9 linear dependencies by Conway et al since one of the terms, again (abc), is not a moonshine function. But just like in Part 1, the (bca) term is the McKay-Thompson series of class 24B (A212771).

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