A127862 Coefficient of modular form for elliptic curve "1323m1": y^2 + y = x^3 - 2 divided by q in powers of q^3.

1, -2, 0, 0, -2, 4, 7, 0, -5, 0, -11, 0, -10, 0, -13, 0, 0, 4, 0, 0, 13, -8, -16, 0, 7, -14, -4, 0, 0, 0, 0, 0, -5, 10, -20, 0, -19, 0, 0, 0, -11, 22, -1, 0, 0, 0, 16, 0, 0, 20, 23, 0, -14, 0, 17, 0, -9, 26, 0, 0, 7, 0, 0, 0, 2, 0, -17, 0, 0, -8, 29, 0, 0, 0, 28, 0, -29, 0, 0, 0, 31, -26, -14, 0, 0, 16, 0, 0, 0, 32, 1

Offset

0,2

Links

Robin Visser, Table of n, a(n) for n = 0..10000

LMFDB, Elliptic Curve 1323.j1 (Cremona label 1323m1).

Formula

a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = b(7^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p == 1 (mod 3) where b(p) = -Sum_{x=0..p-1} Kronecker(4*x^3 - 7, p).

a(4*n + 3) = a(7*n + 2) = 0.

Example

G.f. = 1 - 2*x - 2*x^4 + 4*x^5 + 7*x^6 - 5*x^8 - 11*x^10 - 10*x^12 + ...
G.f. = q - 2*q^4 - 2*q^13 + 4*q^16 + 7*q^19 - 5*q^25 - 11*q^31 + ...

Prog

(PARI) {a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 1, 0, -2], 1), 3*n + 1))};
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3 || p==7, 0, a0 = 1; a1 = y = -sum( x=0, p-1, kronecker( 4*x^3 - 7, p)); for( i=2, e, x = y*a1 - p*a0; a0 = a1; a1 = x); a1)))};
(Magma) qExpansion( ModularForm( EllipticCurve( [0, 0, 1, 0, -2])), 274); /* Michael Somos, Sep 07 2018 */
(Sage)
def a(n):
    return EllipticCurve("1323m1").an(3*n+1)  # Robin Visser, Jan 03 2024

Crossrefs

Sequence in context: A325190 A141416 A176787 * A342223 A223142 A244522

Adjacent sequences: A127859 A127860 A127861 * A127863 A127864 A127865

Keyword

sign

Author

Michael Somos, Feb 03 2007

Status

approved