A247198 Coefficients of L-series for elliptic curve "26a3": y^2 + x * y - y = x^3 + x or y^2 + x*y + y = x^3.

1, -1, 1, 1, -3, -1, -1, -1, -2, 3, 6, 1, 1, 1, -3, 1, -3, 2, 2, -3, -1, -6, 0, -1, 4, -1, -5, -1, 6, 3, -4, -1, 6, 3, 3, -2, -7, -2, 1, 3, 0, 1, -1, 6, 6, 0, 3, 1, -6, -4, -3, 1, 0, 5, -18, 1, 2, -6, -6, -3, 8, 4, 2, 1, -3, -6, 14, -3, 0, -3, -3, 2, 2, 7, 4

Offset: 1

Author

Michael Somos, Nov 24 2014

Keywords

• sign
• mult

Examples

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

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• LMFDB, Elliptic curve with LMFDB label 26.a3 (Cremona label 26a3)

Programs

Magma

A := Basis( CuspForms( Gamma0(26), 2), 76); A[1] - A[2];

PARI

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

PARI

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

Sage

def a(n):
    return EllipticCurve("26a3").an(n)  # Robin Visser, Sep 30 2023

Formula

a(n) is multiplicative with a(2^e) = (-1)^e, a(13^e) = 1, 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).