Wednesday, November 6, 2019
Entry 65
For level >119, there are no more moonshine functions with uppercase index (in Atlas notation). Define dk=η(kτ) with the Dedekind eta function η(τ) then for Level 126, a=(d27d29d3d14d18d21),b=(d21d263d2d3d21d126)c=(d214d218d6d7d9d42),d=(d22d2126d1d6d42d63) The quadruple (a,b,c,d) obey a−2=bc−1=d and their ratios are cubes (ab)2(cd)=(d7d9d1d63)3(ab)(cd)2=(d14d18d2d126)3
Entry 64
Define dk=η(kτ) with the Dedekind eta function η(τ) then for Level 42 a=(d21d214d2d3d7d42),b=(d22d27d1d6d14d21)c=(d23d242d1d6d14d21),d=(d26d221d2d3d7d42) The quadruple (a,b,c,d) obey abcd=1a+3=bc+1=d and their ratios are squaresad=(d1d14d6d21)2bc=(d2d7d3d42)2
Entry 63
For certain even levels divisible by 7 such as 7×4,7×6,7×18 or 28,42,126, we may still find an eta quotient a such that a+m=b is also an eta quotient, but only for one integer m. Define dk=η(kτ) with the Dedekind eta function η(τ) and a=(d1d7d4d28),b=(d32d314d1d24d7d228) then a+2=ba+4a+4=(d22d214d1d4d7d28)3b+4b−4=(d1d7d2d14)3 where (a,b) is the McKay-Thompson series of class 28C for the Monster (A161970).
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