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Search: id:A127826
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| A127826 |
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Coefficients of L-series for elliptic curve "256a1": y^2 = x^3 + x^2 - 3*x + 1. |
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+0
1
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| 1, -2, 0, 0, 1, -6, 0, 0, -6, -2, 0, 0, -5, 4, 0, 0, 12, 0, 0, 0, 6, 10, 0, 0, -7, 12, 0, 0, 4, -6, 0, 0, 0, 14, 0, 0, -2, 10, 0, 0, -11, -18, 0, 0, -18, 0, 0, 0, 10, -6, 0, 0, 0, -6, 0, 0, 18, 0, 0, 0, 25, -12, 0, 0, -20, -18, 0, 0, 6, -22, 0, 0, 0, 14, 0, 0, -6, 0, 0, 0, 0, -2, 0, 0, -13, -2, 0, 0, 12, -18, 0, 0, 0, 36
(list; graph; listen)
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OFFSET
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0,2
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LINKS
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W. Stein, Modular Forms Database.
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FORMULA
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a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1+(-1)^e)/2*(-p^2)^(e/2) if p == 5,7 (mod 8), b(p^e) = b(p)*b(p^(e-1)) -p*b(p^(e-2)) if p == 1,3 (mod 8) and b(p) = 2*x*(-1)^((x%8>4)+(y%4)>0) where p = x^2+2*y^2.
a(4n+2)=a(4n+3)=0.
Expansion of q^(-1/2) * eta(q^8)^8 / (eta(q^4) * eta(q^16))^2 * (eta(q^4) / eta(q^16) - 2 * eta(q^16) / eta(q^4)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = -64 (t/i)^2 f(t) where q = exp(2 pi i t).
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EXAMPLE
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q - 2*q^3 + q^9 - 6*q^11 - 6*q^17 - 2*q^19 - 5*q^25 + 4*q^27 + 12*q^33 + ...
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PROGRAM
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(MAGMA) f := qEigenform(EllipticCurve(CremonaDatabase(), "256a1"), 188); [ Coefficient(f, n) : n in [ k : k in [0..188] | IsOdd(k) ] ] ; /* Klaus Brockhaus, Feb 01 2007 */
(PARI) {a(n)=local(A, p, e, x, y); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p%8>4, if(e%2, 0, (-p)^(e/2)), for(i=1, sqrtint(p\2), if(issquare(p-2*i^2, &x), y=i; break)); a0=1; a1=y=2*x*(-1)^((x%8>4)+(y%4>0)); for(i=2, e, x=y*a1-p*a0; a0=a1; a1=x); a1))))) }
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^(n\4)); polcoeff( eta(x^2+A)^8/ eta(x+A)^2/ eta(x^4+A)^2* ((n%4==0)*eta(x+A)/eta(x^4+A) -(n%4==1)*2*eta(x^4+A)/eta(x+A)), n\4))}
(PARI) {a(n) = ellak( ellinit([0, 1, 0, -3, 1], 1), 2*n + 1)}
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CROSSREFS
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Convolution of A138515(q^4) and A112172.
Sequence in context: A122890 A138497 A113129 this_sequence A109983 A093492 A128771
Adjacent sequences: A127823 A127824 A127825 this_sequence A127827 A127828 A127829
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Jan 30 2007
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