A127863 Coefficients of modular form for elliptic curve "243b1": y^2 + y = x^3 + 2 divided by q in powers of q^3.

1, -2, 5, 0, 2, 4, 8, 0, -5, -10, -7, 0, -1, 0, -13, 0, 18, -4, 0, 0, -1, -8, 5, 0, -7, -16, -4, 0, 0, 0, 10, 0, 14, 10, -13, 0, 17, 20, 0, 0, -11, 14, -19, 0, 40, 0, -7, 0, 0, 2, -19, 0, 11, 0, 17, 0, -9, 26, -25, 0, -19, 0, 0, 0, 23, -36, -28, 0, 0, 8, -16, 0, -35, 0, 5, 0, 29, 0, 0, 0, -31, 2, 16, 0, 0, 16, -5, 0, 0, -10

Offset

0,2

Links

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

LMFDB, Elliptic Curve 243.b2 (Cremona label 243b1).

Formula

a(n) = b(3*n + 1) where b(n) is multiplicative with b(3^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 + 9, p).

a(4*n + 3) = 0.

Example

G.f. = 1 - 2*x + 5*x^2 + 2*x^4 + 4*x^5 + 8*x^6 - 5*x^8 - 10*x^9 - 7*x^10 + ...
G.f. = q - 2*q^4 + 5*q^7 + 2*q^13 + 4*q^16 + 8*q^19 - 5*q^25 - 10*q^28 - ...

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, 0, a0 = 1; a1 = y = -sum( x=0, p-1, kronecker( 4*x^3 + 9, p)); for( i=2, e, x = y*a1 - p*a0; a0 = a1; a1 = x); a1))) };
(Magma) qExpansion( ModularForm( EllipticCurve( [0, 0, 1, 0, 2])), 270); /* Michael Somos, Sep 07 2018 */
(SageMath)
def a(n):
    return EllipticCurve("243b1").an(3*n+1)  # Robin Visser, Jan 03 2024

Crossrefs

Sequence in context: A343605 A296047 A307237 * A006891 A054675 A346191

Adjacent sequences: A127860 A127861 A127862 * A127864 A127865 A127866

Keyword

sign

Author

Michael Somos, Feb 03 2007

Status

approved