A122777 Coefficients of L-series for elliptic curve "30a1": y^2 + x * y + y = x^3 + x + 2.

1, -1, 1, 1, -1, -1, -4, -1, 1, 1, 0, 1, 2, 4, -1, 1, 6, -1, -4, -1, -4, 0, 0, -1, 1, -2, 1, -4, -6, 1, 8, -1, 0, -6, 4, 1, 2, 4, 2, 1, -6, 4, -4, 0, -1, 0, 0, 1, 9, -1, 6, 2, -6, -1, 0, 4, -4, 6, 0, -1, -10, -8, -4, 1, -2, 0, -4, 6, 0, -4, 0, -1, 2, -2, 1, -4, 0, -2, 8, -1, 1, 6, 12, -4, -6, 4, -6, 0, 18, 1, -8, 0, 8, 0, 4, -1, 2, -9, 0, 1, 18, -6, -4

Offset

1,7

Links

G. C. Greubel, Table of n, a(n) for n = 1..1500

Formula

Expansion of eta(q^3) * eta(q^5) * eta(q^6) * eta(q^10) - eta(q) * eta(q^2) * eta(q^15) * eta(q^30) in powers of q.

G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(5*k)) * (1 - x^(6*k)) * (1 - x^(10*k)) - x^2 * Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(15*k)) * (1 - x^(30*k)).

a(n) = A122779(2*n). a(2*n) = - a(n). a(3*n) = a(n). - Michael Somos, Oct 28 2014

Example

G.f. = q - q^2 + q^3 + q^4 - q^5 - q^6 - 4*q^7 - q^8 + q^9 + q^10 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[eta[q^3]* eta[q^5]*eta[q^6]*eta[q^10] - eta[q]*eta[q^2]*eta[q^15]*eta[q^30], {q, 0, 75}], q]; Table[a[[n]], {n, 2, 50}] (* G. C. Greubel, Jul 18 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^5 + A) * eta(x^6 + A) * eta(x^10 + A) - eta(x + A) * eta(x^2 + A) * eta(x^15 + A) * eta(x^30 + A) * x, n))};
(PARI) {a(n) = local(A); if( n<1, 0, n*=2; n--; A = x * O(x^n); A = eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)); A = A * subst(A + x * O(x^(n\5)), x, x^5); polcoeff(A, n))};
(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==2 || p==5, (-1)^e, if( p==3, 1, a1 = y = - sum(x=0, p-1, kronecker( 6*x^3 + x^2 + 4*x + 4, p)); a0 = 1; for(i=2, e, x = y * a1 - p * a0; a0 = a1; a1 = x); a1)))))};
(Magma) A := Basis( CuspForms( Gamma0(30), 2), 104); A[1] - A[2] + A[3];  /* Michael Somos, Oct 28 2014 */

Crossrefs

Cf. A122779.

Sequence in context: A016684 A324564 A276974 * A103524 A110916 A185058

Adjacent sequences: A122774 A122775 A122776 * A122778 A122779 A122780

Keyword

sign,mult

Author

Michael Somos, Sep 10 2006

Status

approved