OFFSET
0,3
LINKS
Robin Visser, Table of n, a(n) for n = 0..10000
FORMULA
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 96 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(n) = b(2*n + 1) where b() is multiplicative with b(3^e) = (-1)^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p>3, where b(p) = p minus number of points of elliptic curve modulo p.
EXAMPLE
G.f. = 1 - x + 2*x^2 + 4*x^3 + x^4 - 4*x^5 - 2*x^6 - 2*x^7 - 6*x^8 + ...
G.f. = q - q^3 + 2*q^5 + 4*q^7 + q^9 - 4*q^11 - 2*q^13 - 2*q^15 - 6*q^17 + ...
MATHEMATICA
a[ n_] := Module[{x, y, p, e, a0, a1}, If[n < 1, Boole[n == 0], Times @@ ( If[# == 3, (-1)^#2, p = #; e = #2; a0 = 1; a1 = y = -Sum[KroneckerSymbol[x^3 - x^2 - 2*x, p], {x, p}]; Do[x = y a1 - p a0; a0 = a1; a1 = x, e - 1]; a1] & @@@ FactorInteger@(2 n + 1) )]];
PROG
(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; my(A = elltaniyama(ellinit([0, -1, 0, -2, 0]), n)); polcoeff( x * deriv(A[1]) / (2*A[2]), n))};
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, n==0, A = factor(2*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; a0=1; if( p==3, (-1)^e, a1=y = -sum( x=1, p, kronecker(x^3 - x^2 - 2*x, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))};
(Magma) qExpansion( ModularForm( EllipticCurve( [0, -1, 0, -2, 0])), 70);
(Magma) A := Basis( CuspForms( Gamma0(96), 2), 70); A[1] - A[3] + 2*A[5] + 4*A[7] + A[8] - 4*A[9];
(Sage)
def a(n):
return EllipticCurve("96b1").an(2*n+1) # Robin Visser, Jan 03 2024
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 04 2018
STATUS
approved