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Wednesday, November 6, 2019

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.

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