A251913 Coefficients of L-series for elliptic curve "26b1": y^2 + x * y - 2*y = x^3 + 2*x^2.

1, 1, -3, 1, -1, -3, 1, 1, 6, -1, -2, -3, -1, 1, 3, 1, -3, 6, 6, -1, -3, -2, -4, -3, -4, -1, -9, 1, 2, 3, 4, 1, 6, -3, -1, 6, 3, 6, 3, -1, 0, -3, -5, -2, -6, -4, 13, -3, -6, -4, 9, -1, 12, -9, 2, 1, -18, 2, -10, 3, -8, 4, 6, 1, 1, 6, -2, -3, 12, -1, -5, 6

Offset

1,3

Links

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

LMFDB, Elliptic curve with LMFDB label 26.b2 (Cremona label 26b1)

Formula

a(n) is multiplicative with a(2^e) = 1, a(13^e) = (-1)^e, else a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p+1 minus number of points of elliptic curve modulo p including point at infinity.

G.f. is a period 1 Fourier series which satisfies f(-1 / (26 t)) = 26 (t / i)^2 f(t) where q = exp(2 Pi i t).

Example

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

Prog

(PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, 2, -2, 0, 0], 1), n))};
(Sage) A = ModularForms( Gamma0(26), 2, prec=72).basis(); A[0] + A[1];
(Magma) A := Basis( CuspForms( Gamma0(26), 2), 72); A[1] + A[2];

Crossrefs

Cf. A247198.

Sequence in context: A325982 A202338 A139002 * A285012 A139378 A038500

Adjacent sequences: A251910 A251911 A251912 * A251914 A251915 A251916

Keyword

sign,mult

Author

Michael Somos, Dec 10 2014

Status

approved