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Wednesday, November 6, 2019

Levels 126 & 252

I. Non-moonshine functions. There are no moonshine functions with uppercase index (in Atlas notation) for level >119. However, surprisingly we can still find trinomial identities for level 144 (some a consequence of level 72) and as high as level 252 (a consequence of level 126). For the latter, define the two pairs of eta quotients, a(τ)=(d27d29d3d14d18d21),b(τ)=(d21d263d2d3d21d126)c(τ)=(d214d218d6d7d9d42),d(τ)=(d22d2126d1d6d42d63)Given one moonshine function of level 21, namely j21B(τ)=(d1d3d7d21) and define, e(τ)=j21B(τ)j21B(3τ)=(d1d63d7d9) then ratios of the pairs are simply, ab=e(2τ)e2(τ)cd=e(τ)e2(2τ) They obey,a(τ)2=b(τ)c(τ)1=d(τ) a(12+τ)2=b(12+τ)c(12+τ)1=d(12+τ) which is the pair of trinomial identities each for level 126 and level 252, respectively, and the latter seems to be the highest known.

Levels 42 & 84

I. Moonshine functions. The four functions for level 42,
j42A(τ)=(j42B+1j42B)2j42B(τ)=(d1d6d14d21d2d3d7d42)2j42C(τ)=3j14B(3τ)=(d3d21d6d42)j42D(τ)=(d2d6d7d21d1d3d14d42) and the three functions for level 84,
j84A(τ)=j42A(2τ)j84B(τ)=j42B(2τ)j84C(τ)=3j28B(3τ)=(d26d242d3d12d21d84)
II. Non-moonshine functions. Define the two pairs of eta quotients, a(τ)=(d21d214d2d3d7d42),b(τ)=(d22d27d1d6d14d21)c(τ)=(d23d242d1d6d14d21),d(τ)=(d26d221d2d3d7d42) Interestingly, these satisfy a(τ)b(τ)c(τ)d(τ)=1 And in terms of the moonshine functions,
ab=j42Bj42Ddc=j42Bj42D They obey, a(τ)+3=b(τ)c(τ)+1=d(τ) a(12+τ)+3=b(12+τ)c(12+τ)+1=d(12+τ) which is the pair of trinomial identities each for level 42 and level 84, respectively.

Levels 14 & 28

For certain even levels divisible by 7, the situation is now different. We can still find an eta quotient a such that a+m=b is also an eta quotient, but only for one rational m.

I. Moonshine functions: Define a(τ)=(d1d7d4d28),b(τ)=(d32d314d1d24d7d228) then a+2=b or, (d1d7d4d28)+2=(d32d314d1d24d7d228) for the unique m=2. This is the single trinomial identity of level 28. First, we have the 3 moonshine functions of level 14,
j14A(τ)=(j14C+1j14C)2j14B(τ)=(d1d7d2d14)3=b+4b4j14C(τ)=(d2d7d1d14)4 and the 4 moonshine functions of level 28,
j28A(τ)=j14A(2τ)=j28D(τ)+1j28D(τ)j28B(τ)=(d22d214d1d4d7d28)3=a+4a+4j28C(τ)=d1d7d4d28=aj28D(τ)=j14C(2τ)=(d4d14d2d28)2 Note that j14B(12+τ)=j28B(τ).

Summary

(Under construction)

Level 80

(Under construction.)