|
| |
|
|
A127862
|
|
Coefficient of L-series for elliptic curve "1323m1": y^2 + y = x^3 - 2.
|
|
0
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
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
| 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) = local(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; 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)))) }
|
|
|
CROSSREFS
| Sequence in context: A028641 A141416 A176787 * A024690 A200728 A066209
Adjacent sequences: A127859 A127860 A127861 * A127863 A127864 A127865
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Michael Somos, Feb 03 2007
|
| |
|
|
