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A341567
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Fourier coefficients of the modular form (1/t_{6a}) * (1-6*sqrt(-3)/t_{6a}) * (1-12*sqrt(-3)/t_{6a}) * F_{6a}^12.
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0
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1, -3, -810, -14848, -123111, -544644, -1362010, -2330370, -3239838, 2585060, 15441024, 4575528, 56803975, 51302133, -76460274, -7843816, 58087692, -680814720, 178532126, -82694130, -573777270, 317066108, 1870315110, -1107862272, 2815890921, 240200154, 1340122806
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OFFSET
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0,2
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COMMENTS
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Here, F_{6a} is the hypergeometric function F(1/3, 1/2; 1; 12*sqrt(-3)/t_{6a}). The definition given on page 23 in the linked manuscript has a minor typo where "t_{3A}" should be "t_{6a}". - Robin Visser, Jul 23 2023
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LINKS
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PROG
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(Sage)
def a(n):
eta = x^(1/24)*product([(1 - x^k) for k in range(1, 2*n+1)])
t6a = ((eta(x=x^2)/eta(x=x^6))^6
- 27*(eta(x=x^6)/eta(x=x^2))^6)(x=sqrt(x)) + 6*sqrt(-3)
F6a = sum([rising_factorial(1/3, k)*rising_factorial(1/2, k)/
(rising_factorial(1, k)^2)*((12*sqrt(-3))/t6a)^k for k in range(2*n+1)])
f = (1/t6a)*(1-6*sqrt(-3)/t6a)*(1-12*sqrt(-3)/t6a)*F6a^12
return f.taylor(x, 0, n+1).coefficients()[n][0] # Robin Visser, Jul 23 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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