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Wednesday, May 28, 2025

Entry 111

Given q=e2πiτ and define the McKay-Thompson series of Class 10E for Monster (A138516) j10E(τ)=η(2τ)η5(5τ)η(τ)η5(10τ)+1=(η2(2τ)η(5τ)η(τ)η2(10τ))2 On a hunch, I decided to test this since it seems similar to Class 6E of the previous entry. It turns out j10E(τ) is also product of fundamental units Un for appropriate τ. (Here is a sample Wolfram calculation for U5.) Let d=20m with class number h(d)=4 for even m=6,14,26,38 and odd m=17j10E(6/52)=U2U10j10E(14/52)=U32U10j10E(26/52)=U313U65j10E(38/52)=U42U95j10E(1+17/52)=U65U17

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