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Search: id:A138661
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| A138661 |
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Expansion of a level 11 weight 7 modular form in powers of q. |
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1
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| 1, 0, 10, 64, 74, 0, 0, 0, -629, 0, -1331, 640, 0, 0, 740, 4096, 0, 0, 0, 4736, 0, 0, -12670, 0, -10149, 0, -13580, 0, 0, 0, 56018, 0, -13310, 0, 0, -40256, 87050, 0, 0, 0, 0, 0, 0, -85184, -46546, 0, -206350, 40960, 117649, 0, 0, 0, 246890, 0, -98494, 0, 0, 0, 107642, 47360, 0
(list; graph; listen)
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OFFSET
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1,3
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LINKS
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W. A. Stein, Modular Forms Database.
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FORMULA
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a(4*n + 2) = 0. a(11*n + 2) = a(11*n + 6) = a(11*n + 7) = a(11*n + 8) = a(11*n + 10) = 0.
a(n) is multiplicative with a(11^e) = (-1331)^e, a(p^e) = p^(3*e) * (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = a(p) * a(p^(e-1)) - p^6 * a(p^(e-2)) if p == 1, 3, 4, 5, 9 (mod 11) where a(p) = y^6 - 6*p*y^4 + 9*p^2*y^2 - 2*p^3 and 4 * p = y^2 + 11 * x^2.
G.f. is period 1 Fourier series which satisfies f(-1/ (11 t)) = 11^(7/2) (t/i)^7 f(t) where q = exp(2 pi i t).
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EXAMPLE
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q + 10*q^3 + 64*q^4 + 74*q^5 - 629*q^9 - 1331*q^11 + 640*q^12 + 740*q^15 + ...
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PROGRAM
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(PARI) {a(n) = local(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==11, (-1331)^e, if( kronecker(-11, p)==-1, if(e%2, 0, (p^3)^e), for(x=1, sqrtint(4*p\11), if( issquare(4*p - 11*x^2, &y), break)); y = y^6 - 6*p*y^4 + 9*p^2*y^2 - 2*p^3; a0=1; a1=y; for(i=2, e, x = y * a1 - p^6 * a0; a0=a1; a1=x); a1))))) }
(PARI) {a(n) = local(A, F1, F2, G1); if( n<1, 0, A = x * O(x^n); F1 = x * (eta(x + A) * eta(x^11 + A))^2; F2 = x * eta(x^2 + A) * eta(x^22 + A); G1 = (F1 + 4 * F2^2 + 8 * x^4 * (eta(x^4 + A) * eta(x^44 + A))^2) / F2; polcoeff( G1 * F1 * (G1^4 - 8*G1^2*F1 + 7*F1^2), n))}
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CROSSREFS
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Cf. A129522, A065099.
Sequence in context: A046638 A101467 A162473 this_sequence A036426 A055855 A061183
Adjacent sequences: A138658 A138659 A138660 this_sequence A138662 A138663 A138664
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Mar 25 2008
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