Level 6. Define dk=η(kτ) with Dedekind eta function η(kτ) and the McKay-Thompson series of class 6B for the Monster.
j6B(τ)=(d2d3d1d6)12
Example. We select d=12m with class number h(−d)=4 and find
m=10,14,26,34m=7,11,19,31,59 such that the following are special quadratic irrationals j6B(12√−10/3)=U125=(1+√52)12j6B(12√−14/3)=U214=(15+4√14)2j6B(12√−26/3)=U426=(5+√26)4j6B(12√−34/3)=U122=(1+√2)12
j6B(1+√−7/32)=−U321=−(5+√212)3j6B(1+√−11/32)=−U211=−(10+3√11)2j6B(1+√−19/32)=−U63=−(2+√3)6j6B(1+√−31/32)=−U393=−(29+3√932)3j6B(1+√−59/32)=−U259=−(530+69√59)2
since they are fundamental units Un. Note the integer (√−(530+69√59)2−1/√−(530+69√59)2)2=−10602 and similarly for the others as discussed in the previous entry.
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