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A128263
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Coefficients of L-series for elliptic curve "17a4": y^2 + x*y + y = x^3 - x^2 - x or y^2 + x*y - y = x^3 - x^2.
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1
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1, -1, 0, -1, -2, 0, 4, 3, -3, 2, 0, 0, -2, -4, 0, -1, 1, 3, -4, 2, 0, 0, 4, 0, -1, 2, 0, -4, 6, 0, 4, -5, 0, -1, -8, 3, -2, 4, 0, -6, -6, 0, 4, 0, 6, -4, 0, 0, 9, 1, 0, 2, 6, 0, 0, 12, 0, -6, -12, 0, -10, -4, -12, 7, 4, 0, 4, -1, 0, 8, -4, -9, -6, 2, 0, 4, 0, 0, 12, 2, 9, 6, -4, 0, -2, -4, 0, 0, 10, -6, -8, -4, 0, 0, 8, 0, 2, -9, 0, 1, -10, 0
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OFFSET
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1,5
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COMMENTS
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Unique cusp form of weight 2 for congruence group Gamma_0(17). - Michael Somos, Aug 11 2011
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LINKS
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FORMULA
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a(n) is multiplicative with a(17^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p minus number of points of elliptic curve modulo p.
G.f. is a period 1 Fourier series which satisfies f(-1 / (17 t)) = 17 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(9*n) = -3 * a(n). a(9*n + 3) = a(9*n + 6) = 0.
Expansion of q * A(q) * B(q^17) - q^5 * A(q^17) * B(q) where A(), B() are the g.f. for A143379, A143378 respectively. - Michael Somos, Jan 01 2009
Expansion of eta(q) * eta(q^4)^2 * eta(q^34)^5 / (eta(q^2) * eta(q^17) * eta(q^68)^2) - eta(q^2)^5 * eta(q^17) * eta(q^68)^2 / (eta(q) * eta(q^4)^2 * eta(q^34)) in powers of q. - Michael Somos, Jan 01 2009
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EXAMPLE
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G.f. = q - q^2 - q^4 - 2*q^5 + 4*q^7 + 3*q^8 - 3*q^9 + 2*q^10 - 2*q^13 - ...
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PROG
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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==17, 1, a0=1; a1 = y = -if( p==2, 1, sum( x=0, p-1, kronecker( 4*x^3 - 3*x^2 - 2*x + 1, p))); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1))))};
(PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, -1, 1, -1, 0], 1), n))};
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^34 + A)^5 / (eta(x^2 + A) * eta(x^17 + A) * eta(x^68 + A)^2) - x^4 * eta(x^2 + A)^5 * eta(x^17 + A) * eta(x^68 + A)^2 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^34 + A)), n))}; /* Michael Somos, Jan 01 2009 */
(Sage) CuspForms( Gamma0(17), 2, prec = 100).0; # Michael Somos, Aug 11 2011
(Magma) Basis( CuspForms( Gamma0(17), 2), 10) [1]; /* Michael Somos, May 27 2014 */
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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