OFFSET
0,2
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 / (108 t)) = 108 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p>5, where b(p) = p minus number of points of elliptic curve modulo p.
EXAMPLE
G.f. = 1 + 5*x - 7*x^2 - x^3 - 5*x^4 - 4*x^5 - x^6 + 8*x^7 + ...
G.f. = q + 5*q^7 - 7*q^13 - q^19 - 5*q^25 - 4*q^31 - q^37 + ...
MATHEMATICA
a[ n_] := Module[ {x, y, p, e}, If[ n < 1, Boole[n == 0], Times @@ ( If[ # < 5, 0, {p, e} = {##}; p^(e/2) ChebyshevU[ e, -Sum[ KroneckerSymbol[ k^3 + 4, p], {k, p}] *p^(-1/2) / 2 ]] & @@@ FactorInteger@(6 n + 1) )]];
PROG
(PARI) {a(n) = if( n<0, 0, n = 6*n + 1; ellan(ellinit([0, 0, 0, 0, 4]), n)[n])};
(PARI) {a(n) = my(A, p, e, y='y); if( n<0, 0, A = factor(6*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, substpol( polchebyshev(e, 2, -1/2/y * sum(k=1, p, kronecker(k^3 + 4, p))) * y^e, y^2, p))))};
(Magma) qExpansion( ModularForm( EllipticCurve( [0, 0, 0, 0, 4])), 386);
(Magma) A := Basis( CuspForms( Gamma0(108), 2), 386); A[1] + 5*A[6] - 7*A[9] - A[10];
(Sage)
def a(n):
return EllipticCurve("108a1").an(6*n+1) # Robin Visser, Jan 03 2024
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 24 2018
STATUS
approved