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Search: id:A129666
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| A129666 |
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Expansion of unique cusp form of weight 4 level 7 in powers of q. |
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+0
1
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| 1, -1, -2, -7, 16, 2, -7, 15, -23, -16, -8, 14, 28, 7, -32, 41, 54, 23, -110, -112, 14, 8, 48, -30, 131, -28, 100, 49, -110, 32, 12, -161, 16, -54, -112, 161, -246, 110, -56, 240, 182, -14, 128, 56, -368, -48, 324, -82, 49, -131, -108, -196, -162, -100, -128, -105, 220, 110, 810, 224, -488, -12
(list; graph; listen)
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OFFSET
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1,3
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REFERENCES
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H. Rosson, G. Tornaria, Central values of quadratic twists for a modular form of weight 4, pp. 315-321 of J. B. Conrey, et. al.,ed., Ranks of Elliptic Curves and Random Matrix Theory, Cambridge University Press, 2007..
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FORMULA
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Expansion of q * phi(-q)^3 * psi(q) * phi(-q^7)^3 * psi(q^7) + 4*q^2 * (phi(-q) * psi(q) * phi(-q^7) * psi(q^7))^2 in powers of q.
Expansion of ((eta(q) * eta(q^7))^3 +4 * (eta(q^2) * eta(q^14))^3) * (eta(q) * eta(q^7))^2/ (eta(q^2) * eta(q^14)) in powers of q.
a(n) is multiplicative with a(7^e) = (-7)^e, a(p^e) = a(p)*a(p^(e-1)) -p^3*a(p^(e-2)).
G.f. is Fourier series of a weight 4 level 7 cusp form. f(-1/ (7 t)) = 49 t^4 f(t) where q = exp(2 pi i t) .
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 + 2*u*v + 16*u*w + 12*v^2 + 32*v*w + 256*w^2)*(-v^3 + 2*w*u*v + w*u^2 + 16*w^2*u) +2*v^5.
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EXAMPLE
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q - q^2 - 2*q^3 - 7*q^4 + 16*q^5 + 2*q^6 - 7*q^7 + 15*q^8 - 23*q^9 - ...
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PROGRAM
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(PARI) {a(n)= local(A, A1, A2); if(n<1, 0, n--; A= x*O(x^n); A1= eta(x+A)* eta(x^7+A); A2= eta(x^2+A)* eta(x^14+A); polcoeff( (A1^3 +4*x*A2^3)* A1^2/A2, n))}
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CROSSREFS
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Convolution of A002652 and A002656.
Adjacent sequences: A129663 A129664 A129665 this_sequence A129667 A129668 A129669
Sequence in context: A050612 A120110 A047694 this_sequence A135781 A041573 A041341
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Apr 27 2007
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