|
| |
|
|
A127863
|
|
Coefficients of L-series for elliptic curve "243b1": y^2 + y = x^3 + 2.
|
|
0
| |
|
|
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
(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) = 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
| 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) = 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, 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)))) }
|
|
|
CROSSREFS
| Sequence in context: A118349 A011183 A005671 * A006891 A054675 A136209
Adjacent sequences: A127860 A127861 A127862 * A127864 A127865 A127866
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Michael Somos, Feb 03 2007
|
| |
|
|
