A256678 Coefficients of L-series for elliptic curve "34a1": y^2 + x*y = x^3 - 3*x + 1.

1, 1, -2, 1, 0, -2, -4, 1, 1, 0, 6, -2, 2, -4, 0, 1, -1, 1, -4, 0, 8, 6, 0, -2, -5, 2, 4, -4, 0, 0, -4, 1, -12, -1, 0, 1, -4, -4, -4, 0, 6, 8, 8, 6, 0, 0, 0, -2, 9, -5, 2, 2, -6, 4, 0, -4, 8, 0, 0, 0, -4, -4, -4, 1, 0, -12, 8, -1, 0, 0, 0, 1, 2, -4, 10, -4

Offset

1,3

Links

Robin Visser, Table of n, a(n) for n = 1..10000

LMFDB, Elliptic curve with LMFDB label 34.a4 (Cremona label 34a1)

Formula

a(n) is multiplicative with a(2^e) = 1, a(17^e) = (-1)^e. For p prime, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) and a(p) = p minus number of points of elliptic curve modulo p.

a(2*n) = a(n).

G.f. is a period 1 Fourier series which satisfies f(-1 / (34 t)) = 34 (t/i)^2 f(t) where q = exp(2 Pi i t).

Example

G.f. = q + q^2 - 2*q^3 + q^4 - 2*q^6 - 4*q^7 + q^8 + q^9 + 6*q^11 - 2*q^12 + ...

Prog

(PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, 0, 0, -3, 1], 1), n))};
(Magma) A := Basis( CuspForms( Gamma0(34), 2), 77); A[1] + A[2] - 2*A[3];
(Sage) A = CuspForms( Gamma0(34), 2, prec = 77) . basis(); A[0] + A[1] - 2*A[2];

Crossrefs

Sequence in context: A098542 A320019 A141343 * A066709 A258051 A174026

Adjacent sequences: A256675 A256676 A256677 * A256679 A256680 A256681

Keyword

sign,mult

Author

Michael Somos, Apr 07 2015

Status

approved