Define dk=η(kτ) with the Dedekind eta function η(τ). Then for Level 30
d6d10d3d5(d1d15d2d30)2+1=d1d15d3d5(d6d10d2d30)2d6d10d3d5(d1d15d2d30)2+2=(d3d5d2d30)Expressed as the triple of eta quotients (a,b,c) such that a+1=ba+2=c
abc=(d1d6d10d15d2d3d5d30)3bca=(d3d5d6d10d1d2d15d30)acb=(d1d3d5d15d2d6d10d30)
then (m,n)=(1,2) and
abc=(d1d6d10d15d2d3d5d30)3bca=(d3d5d6d10d1d2d15d30)acb=(d1d3d5d15d2d6d10d30)
where (a,b,c) are the McKay-Thompson series of class 30G (A133098). They obey (abc+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). So this is the highest level.
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