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Thursday, October 31, 2019

Entry 45

Level 12 (Part 2).  For Part 2, one term of the linear dependency is not a moonshine function. Define dk=η(kτ) with the Dedekind eta function η(τ). Then (d24d6d2d212)21=(d1d24d96d32d33d612)(d24d6d2d212)2+3=(d72d3d31d24d6d212) Expressed as the triple of eta quotients (a,b,c) such that a1=ba+3=c so (m,n)=(1,3). Then abc=(d1d24d36d32d3d212)4bca=(d32d36d1d3d24d212)2acb=(d22d3d1d26)4d=(d1d3d2d6)6 where (a,b,c) are still the McKay-Thompson series of class 12I (A187144) but d is the McKay-Thompson series of class 6C (A121666). They obey (16abcbca)2=d d+2a=abc+bca+acb8 in contrast to Part 1 which used the positive sign. Equivalently, (16abc+bca)+2a=abc+bca+acb However, this is not one of the 9 linear dependencies by Conway et al since the (abc) term (which is A193522) is not a moonshine function.

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