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A128263 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. 0

%I

%S 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,

%T 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,

%U -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

%N 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.

%C Unique cusp form of weight 2 for congruence group Gamma_0(17). - _Michael Somos_, Aug 11 2011

%F 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.

%F 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).

%F a(9*n) = -3 * a(n). a(9*n + 3) = a(9*n + 6) = 0.

%F 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

%F 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

%e 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 - ...

%o (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))))};

%o (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, -1, 1, -1, 0], 1), n))};

%o (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 */

%o (Sage) CuspForms( Gamma0(17), 2, prec = 100).0; # _Michael Somos_, Aug 11 2011

%o (MAGMA) Basis( CuspForms( Gamma0(17), 2), 10) [1]; /* _Michael Somos_, May 27 2014 */

%K sign,mult

%O 1,5

%A _Michael Somos_, Feb 21 2007

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Last modified August 11 19:16 EDT 2020. Contains 336428 sequences. (Running on oeis4.)